PRACTICAL  SURVEYING 

FOR 

SURVEYORS'  ASSISTANTS, 
VOCATIONAL,  AND   HIGH    SCHOOLS 


BY 

ERNEST  McCULLOUGH,  C.E.  Ph.D. 

Consulting  Engineer;  Member  of  the  American  Society  of  Civil  Engineers. 

Sometime  Lieut.-Colonel,  United  States  Army,  A.  E.  F.;  Author  of 

"Engineering  as  a  Vocation1';  "Practical  Structural  Design,"  etc. 


229  ILLUSTRATIONS 


SECOND    EDITION,    REVISED 


NEW  YORK 
D.   VAN    NOSTRAND    COMPANY 

EIGHT  WARREN  STREET 
1921 


»» '  .  • 

*•  v : 

.  • 


i\A 


Copyright,  1915, 
By  D.  VAN  NOSTRAND  CO. 

Copyright,  1921, 
By  D.  VAN  NOSTRAND  CO. 


To 
LEWIS  INSTITUTE   (CHICAGO) 

IN   RECOGNITION   OF   THE   WORK 
IT   IS   DOING 


469981 


PREFACE  TO  THE  SECOND  EDITION 

During  the  recent  world  war,  into  which  the  United 
States  entered  in  1917,  thousands  of  men  had  to  be  in- 
structed in  the  principles  of  surveying.  Not  only  did  the 
engineers  need  men  of  non-commissioned  grade  to  serve  as 
instrument-men  and  makers  of  topograpnical  maps,  but  the 
artillery  also  required  them.  In  the  conventionalized 
courses  given  in  the  conventionalized  technical  schools  of 
the  country  it  had  been  assumed  that  surveying  could  be 
taught  only  to  men  who  had  completed,  in  high  school  and 
college,  algebra,  plane  and  solid  geometry,  trigonometry  and 
analytical  geometry.  During  the  war  thousands  of  intelli- 
gent men  whose  mathematical  preparation  did  not  extend 
beyond  the  arithmetic  given  in  grade  schools,  were  trained 
in  a  few  weeks  in  the  use  of  surveying  instruments  and 
received  adequate  instruction  in  the  handling  of  formulas 
used  by  surveyors.  They  were  not  finished  surveyors,  for 
their  training  was  of  a  special  sort,  but  with  the  elements  of 
surveying  given  them  many  were  able  after  the  close  of 
hostilities  to  continue  their  studies.  The  author  benefitted, 
for  this  book,  the  first  edition  of  which  appeared  in  1915, 
was  the  only  available  modern  work  on  surveying  written 
for  men  who  had  not  studied  enough  mathematics,  higher 
than  arithmetic,  to  enable  them  to  read  modern  texts  on 
surveying  written  by  college  and  university  professors. 

The  author  in  1915  was  entering  upon  his  twenty-eighth 
year  of  practice  as  a  civil  engineer,  of  which  sixteen  years 
had  been  spent  in  the  far  western  states  in  the  Rocky 
Mountains  and  along  the  Pacific  Coast.  He  had  served  as  a 
town  and  city  engineer  in  small  and  medium  size  munici- 


VI  PREFACE    TO    SECOND    EDITION 

palities,  as  a  deputy  county  surveyor  and  as  a  United  States 
Deputy  Mineral  Surveyor.  He  had  served  also  as  a  teacher 
of  surveying  in  evening  classes  of  schools  operated  not  for 
profit,  and  numbers  of  boys  and  young  men  appreciated  the 
instruction  given  them,  many  of  whom  later  met  with 
considerable  success  in  surveying  and  civil  engineering  work. 
In  that  year  he  revised  the  lessons  prepared  for  his  pupils 
and  put  them  into  book  form  for  the  benefit  of  others  who 
might  wish  to  study  the  subject  in  a  plainly  written  book. 
That  it  has  sold  well  proves  his  ideas  to  have  been  right. 
The  work  of  the  war  schools  in  surveying  was  additional 
confirmation  that  the  principles  of  surveying  can  be 
taught  to  men  whose  knowledge  of  mathematics  is  meager. 

The  following  specifications  must  be  adhered  to  in  pre- 
paring a  text  book;  (a)  it  must  be  accurate  and  clear,  (b) 
it  must  be  well  balanced,  especially  when  intended  for  men 
who  will  obtain  little,  or  no,  help  from  teachers,  (c)  it  must 
be  readable.  The  author  in  attempting  to  comply  with  these 
specifications  proved  that  a  book  could  be  prepared  which 
did  not  lack  in  vigor  because  of  the  assumption  of  little 
previous  mathematical  preparation  on  the  part  of  the 
student.  It  contains  more  than  is  commonly  given  in 
schools  and  colleges  where  some  instruction  is  given  in 
surveying  as  a  part  of  the  course  in  mathematics,  often- 
times under  instructors  who  have  had  little  or  no  practical 
surveying  experience.  It  contains  more  than  is  required  as 
a  basic  surveying  course  for  non-commissioned  officers  of 
artillery  and  the  corps  of  engineers. 

The  subject  is  here  presented  in  a  logical  manner.  As- 
suming that  the  student  is  familiar  with  common  school 
arithmetic  all  necessary  instruction  in  mathematics  is  given 
step  by  step  as  the  need  is  felt  for  it,  and  not  before.  The 
chapter  on  trigonometry  contains  what  should  be  properly 
considered  the  minimum  a  surveyor  should  possess,  although 


PREFACE    TO    SECOND    EDITION  Vll 

many  surveyors  do  earn  a  living  who  know  only  a  few  of  the 
formulas  there  presented.  The  essentials  of  algebra  were 
placed  in  an  appendix  to  serve  merely  as  a  useful  introduc- 
tion to  the  study  of  algebra.  This  appendix  has  been  highly 
spoken  of  and  the  author  is  gratified  that  it  met  with  the 
approval  of  men  who  are  competent  judges. 

The  author  believes  he  succeeded  in  showing  that  there 
is  more  to  surveying  than  the  ability  to  solve  triangles  and 
read  a  vernier,  important  as  these  items  are.  The  insistence 
from  the  first  chapter  on  the  large  part  played  by  unescap- 
able  errors  and  the  action  of  courts  and  juries  in  passing 
upon  the  work  of  surveyors,  gives  a  view  of  the  subject  that 
is  seldom  obtained  until  after  the  class  room  is  a  memory 
and  actual  work  begins.  Land  surveying  is  emphasized,  for 
land  survey  methods  underlie  all  work  done  by  instru- 
mentmen.  Enough  is  given  of  engineering  surveying  to 
help  local  surveyors  over  many  hard  places.  The  work  is 
complete  in  everything  with  which  the  average  surveyor 
will  have  to  deal:  titles  and  prices  being  given  of  books 
which  deal  fully  with  special  branches  of  survey  work. 
The  prices  given  are  those  of  before  the  war  and  may  not 
be  in  all  cases  the  prices  of  to-day.  Some  however  have  not 
changed  and  changes  may  go  one  way  as  easily  as  another, 
so  it  was  thought  best  to  retain  the  original  prices  in  this 
second  edition. 

The  first  class  surveyor  the  author  has  found  is  a  respected 
person  in  his  community,  and  rightly  so.  Every  reader  of 
this  book  is  counselled  to  buy  and  read  "Boundaries  and 
Landmarks,"  by  A.  C.  Mulford,  to  obtain  the  point  of  view 
of  the  successful  surveyor  who  is  proud  of  his  calling,  the 
most  ancient  of  the  learned  arts  and  sciences.  To  read  that 
book  gives  something  all  young  men  need  and  should 
appreciate,  the  nearest  possible  equivalent  to  association 
with  an  elderly,  well  read,  kindly  gentleman  of  broad 


Vlll  PREFACE    TO    SECOND    EDITION 

experience  in  his  calling.  The  reproach  under  which  sur- 
veyors suffer  is  that  many  have  picked  up  the  business  in  a 
haphazard  way  for  lack  of  comprehendible  texts,  the 
majority  of  graduates  in  engineering  preferring  to  engage 
in  engineering  work  and  not  liking  the  work  of  the  surveyor. 
This  leaves  the  field  then  to  self-tutored  men  working  with- 
out proper  instruction,  or  at  least  instruction  not  as  com- 
plete as  it  should  be.  That  men  in  this  class  are  willing  to 
study  when  simply  written  books  are  placed  in  their  hands 
is  evidenced  by  the  reception  given  to  the  first  edition  of 
this  work. 

Opportunity  we  are  told  is  half  of  life.  An  old  engineer 
gave  the  writer  in  1888  the  following  recipe  for  success: 

Opportunity %  part 

Common  sense J/£  part 

Special  training J^j  part 

which  implies,  that,  given  opportunity,  success  is  due  to  a 
mixture  of  two-thirds  natural  ability  and  one-third  special 
training.  This  book  is  intended  to  give  the  special  training, 
so  far  as  it  may  be  given  in  books,  to  self-tutored  young 
men  of  natural  ability  who  like  surveying  and  to  whom  the 
opportunity  comes  to  obtain  work  in  the  office  or  field  with 
practicing  surveyors. 

Thousands  of  men  have  contributed  to  the  advancement 
of  the  science  and  art  of  surveying,  and  it  is  therefore  not 
possible  to  give  credit  to  past  writers,  except  to  say  that  in 
preparing  this  book  the  literature  of  the  subject  has  been 
well  searched.  Credit  has  been  given  where  but  one  author 
is  involved,  matters  discussed  by  several  writers,  in  this 
subject  which  has  filled  hundreds  of  volumes,  being  assumed 
to  be  common  knowledge.  The  author  believes  he  contrib- 
uted nothing  original,  for  even  things  which  he  learned 
from  experience  may  not  have  been  original  even  though 


PREFACE    TO    SECOND    EDITION  IX 

heretofore  not  mentioned  In  text  books,  but  he  is  gratified 
that  many  purchasers  took  the  trouble  to  write  that  they 
liked  his  manner  of  presentation  and  found  the  book  of 
considerable  value.  One  of  the  most  delightful  incidents  of 
this  nature  was  a  meeting  on  the  British  front  in  the  winter 
of  1917-8,  with  an  English  officer  who  was  studying  the 
book  preparatory  to  following  the  profession  of  surveying  in 
Africa,  when  the  war  should  be  over.  He  went  there  in 
1919,  fortunately  none  the  worse  for  wounds,  and  the 
acquaintance  thus  begun  promises  to  endure  as  a  warm 
friendship. 

The  second  edition  differs  from  the  first  merely  in  the 
correction  of  all  errors  discovered.  Readers  are  invited  to 
call  attention  to  any  errors  which  may  still  exist,  in  case 
they  discover  any.  Hints  and  suggestions  for  improvements 
are  invited.  It  is  believed  that  six  years  of  sales  which  show 
increases  from  year  to  year,  in  spite  of  the  fact  that  three 
were  years  of  war,  may  be  considered  as  proof  that  the 
book  may  be  now  classed  as  standard. 

THE  AUTHOR. 

New  York  City,  N.  Y.,  September,  1921. 


TABLE    OF    CONTENTS 

CHAPTER  PAGE 

I.   INTRODUCTORY i 

II.   CHAIN  SURVEYING 14 

III.  LEVELING 73 

IV.  COMPASS  SURVEYING 100 

V.  TRIGONOMETRY 148 

VI.  TRANSIT  SURVEYING 210 

VII.   SURVEYING  LAW  AND  PRACTICE 290 

VIII.   ENGINEERING  SURVEYING 33° 

APPENDIX  A.  THE  ESSENTIALS  OF  ALGEBRA 363 


PRACTICAL     SURVEYING 


CHAPTER    I 
INTRODUCTORY 

Gravitation  is  a  natural  force  acting  on  all  material 
bodies  with  the  effect  of  attracting  them  to  each  other. 

Terrestrial  gravitation  (gravity)  is  the  operation  of  the 
law  of  gravitation  so  that  all  bodies  within  the  range  of  its 
influence  tend  to  be  drawn  to  the  center  of  the  earth. 
That  is,  a  straight  line  drawn  from  any  point  on  the  surface 
of  the  earth  in  the  direction  of  its  center  is  said  to  lie  in  the 
direction  of  gravity.  In  many  cases  gravity  means  simply 
weight,  which  is  a  measure  of  the  force  of  gravity  acting 
on  a  body. 

A  cord  having  a  plumb-bob  attached  to  the  lower  end 
points  in  the  direction  of  gravity  and  is  said  to  define  a 
vertical  line. 

Differences  in  elevation  are  measured  on  vertical  lines. 

A  line  forming  a  right  angle  with  a  vertical  line  is  defined 
in  plane  surveying  as  being  level,  or  horizontal. 

A  point  has  position  without  length,  breadth  or  thick- 
ness. A  line  has  position  and  length  without  breadth  or 
thickness.  A  line  is  terminated  by  points  at  the  ends; 
it  may  be  said  to  be  composed  of  a  number  of  points  touch- 
ing each  other;  it  may  be  said  to  be  generated  by  a  point 
changing  position  in  space,  the  first  position  being  the  begin- 
ning of  the  line,  the  final  position  of  the  point  being  the 
other  end,  the  line  between  the  two  points  marking  the 
path  traveled.  This  is  illustrated  every  time  a  line  is 
drawn  with  a  pencil  or  pen. 

A  curved  line  changes  direction  at  each  point  and  a 
straight  line  is  the  shortest  distance  between  two  points. 


;  PRACTICAL   SURVEYING 


A  broken  line  consists  of  a  number  of  short  straight  lines 
joined  end  to  end  and  changing  direction  at  the  junction 
points.  The  changes  in  direction  are  angular,  an  angle 
being  the  amount  of  divergence  between  two  lines  that 
join  or  cross. 

A    surface    has    position,    length    and    breadth    without 
thickness.     A  plane  surface,  or  plane,  is  perfectly  flat  so 


FIG.  i.    Types  of  plumb-bobs  used  by  surveyors. 

that  a  straight  line  may  be  drawn  to  connect  any  two 
points  in  the  plane  and  each  point  in  the  line  will  touch  the 
plane. 

In  plane  surveying  the  portion  of  the  earth's  surface 
measured  is  small  in  comparison  with  the  circumference  of 
the  earth,  so  the  curvature  of  the  earth  is  safely  neglected, 
an  assumption  which  simplifies  operations.  Consequently 
a  horizontal  plane  is  dealt  with. 

In  geodetic  surveying  the  curvature  of  the  earth  is  taken 
into  account,  for  the  operations  are  so  extensive  that 
boundaries  between  nations  are  determined.  The  use  of 
geodetic  methods  is  confined  almost  entirely  to  the  highest 
grade  of  Government  work.  Large  cities  are  surveyed  by 
a  combination  of  geodetic  and  plane  surveying  methods. 


INTRODUCTORY  3 

The  methods  of  plane  surveying  are  used  on  work  of  the 
following  character: 

Land  surveys  to  determine  boundaries  of  fields  or  lots 
and  areas  of  same. 

Canal,  road  and  railway  surveys  to  determine  routes  to 
be  followed  and  the  quantities  of  materials  to  be  moved  in 
forming  the  excavations  and  embankments. 

Construction  surveys  for  the  purpose  of  setting  stakes 
for  the  location  of  buildings,  bridges  and  other  structures; 
computations  of  quantities,  etc. 

Mining  surveys  to  guide  miners  in  driving  tunnels,  shafts 
and  other  workings.  These  are  a  combination  of  land  and 
construction  survey  work. 

Topographical  surveys  made  to  obtain  data  for  maps  on 
which  are  shown  all  natural  physical  characteristics,  such 
as  the  shape  and  heights  of  hills,  extent  of  lowlands,  routes 
of  rivers,  streams,  roads,  etc.,  so  that  improvements  may 
be  planned.  On  extensive  topographical  surveys  a  frame- 
work may  be  used  of  lines  fixed  by  geodetic  methods,  the 
"filling  in"  being  done  by  plane  surveying  methods. 

LAND   SURVEYS 

The  river  Nile  in  Egypt  annually  overflows  its  banks  and 
deposits  upon  the  adjacent  bottom  lands  many  tons  of  silt, 
which,  being  washed  down  from  rich  land  on  the  mountains 
of  the  interior  of  Africa,  makes  the  valley  wonderfully 
fertile.  This  silt  covers  the  stones  and  'posts  set  for  land- 
marks and  swirling  water  often  removes  them,  so  that 
from  the  most  ancient  times  the  boundaries  of  land  in 
Egypt  have  been  surveyed  annually.  The  science  of  geom- 
etry arose  in  consequence  according  to  Diodorus  and  others, 
although  some  authorities  claim  the  science  of  geometry 
was  of  indigenous  growth,  appearing  in  each  country  as 
the  people  reached  a  proper  development.  That  survey- 
ing preceded  geometry  is  indicated  by  the  Latin  name, 
geometria,  from  the  Greek  yeunerpia  (the  measurement 
of  land).  The  first  geometers  were  land  surveyors. 

When  a  portion  of  a  tract  of  land  is  sold  the  seller  and 
buyer  proceed  to  " establish"  the  corners  where  the  angular 
boundary  lines  join  and  a  surveyor  is  employed  to  determine 


4  PRACTICAL  SURVEYING 

the  lengths  and  directions  of  the  lines  between  corners.  A 
description  of  the  corners,  together  with  the  directions  and 
lengths  of  all  lines,  is  written  in  the  deed  conveying  title 
to  the  land  and  sometimes  a  map  is  made.  Such  a  survey 
is  termed  "original,"  for  by  it  the  boundaries  and  marks 
are  first  described. 

In  an  original  survey  the  corners  are  established.  This 
can  never  again  be  done.  If  by  any  chance  the  marks  are 
lost  surveyors  are  employed  to  "re-locate"  if  possible  the 
position  of  the  corners,  but  no  man  can  "re-establish" 
anything  once  established.  A  surveyor,  making  a  re- 
survey,  can  only  say  that  he  has  gone  over  the  lines  to  the 
best  of  his  ability  from  information  given  to  him  and 
believes  he  has  set  marks  .as  nearly  as  possible  where  the 
original  marks  were  placed. 

A  surveyor,  not  being  an  interested  party,  cannot  pre- 
sume to  put  in  a  new  corner  having  the  force  and  effect  of 
an  original  corner.  Some  time  after  he  is  gone  good  evi- 
dence may  be  found  showing  that  the  true  corner  lies  at  a 
considerable  distance  from  the  corner  set  by  him.  The 
owners  being  the  only  parties  having  any  vital  interest  in 
the  matter  may  destroy,  if  they  wish,  all  evidences  of  the 
re-survey  and  abide  by  the  boundaries  originally  fixed. 
The  surveyor  however  should  be  notified  so  he  may  alter 
his  notes.  If  this  is  not  done  trouble  may  arise  in  the 
following  generation. 

Sometimes,  after  several  unsuccessful  attempts  to  re- 
locate a  missing  corner,  the  owners  affected  agree  to  accept 
a  certain  point  in  lieu  of  the  corner  originally  set.  This, 
however,  can  be  done  only  by  mutual  agreement  duly 
recorded  and  it  does  not  constitute  a  "re-establishing"  of 
a  corner,  but  is  the  "establishing"  of  a  new  corner.  If  the 
agreement  is  not  a  mutual  one  between  all  the  parties  at 
interest  and  no  proper  record  is  made  of  it,  a  discovery  at  a 
future  time  of  the  original  corner  may  cause  trouble.  In 
some  states  corners  mutually  held  to  become  fixed  after 
the  lapse  of  a  certain  number  of  years,  but  this  does  not 
act  always  as  a  final  settlement.  In  the  absence  of  records 
it  is  easy  for  some  person  to  deny  that  the  agreement  was 
mutual.  There  may  be  unearthed  records  to  show  that 
the  agreement  was  based  on  fraud,  so  it  is  better  to  have  a 


INTRODUCTORY  5 

carefully  drawn  up  agreement  placed  on  record  than  to 
depend  on  the  statute  of  limitations.  The  fact  a  student 
must  never  forget  is  that  no  surveyor,  by  virtue  of  any 
official  position  he  may  hold,  can  "fix,"  "re-establish,"  or 
"establish,"  any  old  corner. 

A  surveyor  on  an  original  survey  is  employed  to  describe 
the  intentions  of  a  grantor  in  selling  and  of  a  grantee  in  buy- 
ing. A  surveyor  on  a  re-survey  tries  to  retrace  the  lines 
run  by  the  surveyor  on  the  original  survey.  If  the  first 
man  did  his  work  well  no  troubles  should  develop  even 
though  nearly  all  the  original  corner  marks  have  been 
destroyed.  If  the  first  surveyor  made  mistakes  trouble 
cannot  be  avoided  unless  the  second  surveyor  possesses  a 
good  knowledge  of  surveying,  a  good  knowledge  of  court 
decisions  affecting  surveys,  and  plenty  of  common  sense. 

It  is  a  physical  impossibility  to  make  surveys  positively 
free  from  error,  but  by  proceeding  carefully  and  taking  all 
the  time  that  is  necessary  errors  may  be  reduced  to  a  small 
amount  which  will  be  satisfactory  to  all  concerned.  Good 
work  takes  time  and  surveyors  are  generally  paid  by  the 
day.  It  follows  that  the  accuracy  of  surveyors'  work  must 
be  governed  by  the  value  of  the  land  surveyed.  Various 
authors  give  limits  of  accuracy  based  on  experience,  which 
the  author  a  number  of  years  ago  reduced  to  the  following 
rule  : 

The  limit  of  error  in  land  surveys  may  be  expressed  by  a 
fraction  having  a  numerator  of  I  and  a  denominator  equal  to 
one-tenth  the  value  of  the  land  per  acre  expressed  in  cents,  the 
maximum  limit  being  -gkv,  corresponding  to  a  value  of  $50 
per  acre. 

When  a  compass  is  used  to  obtain  directions  and  a  chain 
in  the  hands  of  unskilled  men  is  used  to  measure  distances 
the  error  may  be  as  large  as  I  in  500.  A  very  little  expe- 
rience will  enable  men  to  do  better  work,  so  this  limit  of 
error  should  never  be  considered  satisfactory.  A  limit  of  I 
in  750  should  be  attempted. 

The  limit  of  error  for  land  worth  $100  per  acre  will  be 
i  in  looo  by  the  above  rule  and  for  land  worth  $1000  per 
acre  the  limit  of  error  will  be  I  in  10,000.  An  error  of  I  in 
20,000  is  permissible  in  the  business  district  of  any  large 
city,  except  possibly  in  the  hearts  of  cities  having  a  popula- 


6  PRACTICAL   SURVEYING 

tion  of  more  than  1,000,000  where  a  limit  of  I  in  50,000  may 
be  worth  attempting. 

An  old  legal  maxim  reads,  "  Monuments  govern  courses 
and  courses  govern  distances."  This  arose  from  the  fact 
that  monuments  are  marks  set  by  the  grantor  as  visible 
evidences  of  intention.  The  original  survey  was  an  oper- 
ation performed  to  obtain  data  for  a  record  of  intention. 
It  was  not  considered  that  chaining  was  a  particularly 
skilled  avocation  and  untrained  men  usually  did  such 
work.  It  was  not  uncommon  to  have  the  owner  and  one 
of  his  sons  act  as  chainmen  in  order  to  impress  upon  the 
memory  the  location  of  the  corners.  In  Great  Britain 
"whipping  the  bounds"  has  not  entirely  gone  out  of  style, 
and  this  was  as  common  in  the  early  history  of  America  as 
it  was  in  older  countries.  On  a  certain  day  in  each  year, 
usually  just  after  the  spring  planting,  boys  were  taken 
around  every  piece  of  property  and  whipped  soundly  at 
each  corner  after  the  names  of  the  owners  whose  lands  the 
fences  and  corner  determined,  were  read  to  them.  The 
evidence  painfully  so  secured  was  often  invaluable  in  law- 
suits years  later,  sometimes  when  the  whipped  boys  were 
almost  in  their  second  childhood.  In  the  early  days  chain- 
ing, therefore,  was  considered  to  be  on  a  par  with  the  read- 
ing of  bearings  for  surveyors  were  not  always  well  educated 
and  many  knew  little,  or  nothing,  about  the  variation 
(declination)  of  the  needle.  The  effect  of  local  attraction 
was  generally  ignored ;  compasses  were  not  always  well 
made  and  readings  were  seldom  taken  closer  than  the 
nearest  half  degree  of  bearing. 

Nevertheless,  with  all  the  shortcomings  of  the  angle 
work  the  bearings  were  given  first  consideration  for  they 
at  least  gave  a  close  idea  of  direction,  without  which  the 
most  careful  measurement  is  worthless. 

The  compass  is  not  used  today  for  surveys  where  land 
is  valuable  for  readings  cannot  be  taken  closer  than  one- 
quarter  of  a  degree  with  the  best  made  compass.  When 
skilled  chainmen  are  employed  the  errors  in  chaining  are 
apt  to  be  much  smaller  than  the  errors  in  angle.  Modern 
instruments  are  well  made  and  angles  can  be  read  to  half 
minutes  on  even  the  lower-priced  transits,  while  many 
men  have  instruments  graduated  to  read  to  ten  seconds, 


INTRODUCTORY  7 

twenty  seconds  being  the  usual  degree  of  accuracy  for  en- 
gineering surveys.  Modern  surveyors  are  better  trained 
than  the  surveyors  of  preceding  generations,  so  the  in- 
strument work  leaves  little  to  be  desired  in  point  of  ac- 
curacy. Errors  today  are  most  apt  to  occur  in  the  chaining, 
for  only  by  experience  can  men  be  trained  to  measure 
with  a  chain  or  tape.  Young  surveyors  should  impress 
this  fact  upon  employers  who  want  to  save  expenses,  and 
object  to  paying  a  good  price  to  an  experienced  chainman. 
The  picking  up  of  some  unemployed  laborer  to  do  the 
measuring  is  a  fruitful  cause  of  lawsuits. 

In  reading  angles  the  plate  of  the  instrument  must  be 
level,  and  the  chain  or  tape  must  be  level  and  be  drawn 


' .  '\  H&&  /,T4Wf '     ,  .'  S'  t* 


FIG.  2.    Measuring  on  sloping  ground. 

taut.  The  chainmen  are  carefully  "lined  in"  by  the  in- 
strument man  in  order  that  the  measured  line  may  be  a 
straight,  not  a  broken,  line.  Each  man  has  a  plumb-bob 
on  a  cord  by  means  of  which  he  fixes  each  tape  length,  for 
the  tape  must  be  held  high  enough  to  clear  all  bushes, 
stones,  etc.  On  level  ground  the  tape  is  usually  held 
about  the  height  of  the  waist.  On  sloping  ground  one 
man  holds  his  end  as  close  as  possible  to  the  ground,  the 
other  man  raising  his  end  until  the  tape  is  level,  letting 
the  plumb-bob  line  slide  between  his  fingers  slowly  until  the 
point  of  the  plumb-bob  touches  the  ground.  When  the 


8 


PRACTICAL   SURVEYING 


slope  is  steep  short  measurements  are  taken.  In  meas- 
uring up  or  down  hill  in  short  sections  like  steps  the  addi- 
tive process  is  used  to  avoid  errors.  For 
example  assume  it  is  necessary  to  meas- 
ure in  lengths  of  approximately  20  ft. 
The  head  chainman  first  pulls  the  tape 
(or  chain)  ahead  the  full  length,  then 
returns  to  the  20  ft.  mark  and  sets  a  pin 
while  the  rear  chainman  holds  his  end  on 
the  starting  point.  Then  the  rear  chain- 
man drops  his  end  and  goes  forward  to 
the  head  chainman,  gives  him  a  pin  in 
place  of  the  one  just  set,  holds  at  the 
20  ft.  mark  while  the  head  chainman 
goes  to  the  40  ft.  mark,  when  the  proc- 
ess is  repeated.  In  this  manner  the  full 
length  is  measured  without  any  adding 
being  done  mentally.  When  the  full  tape 
length  is  set  off  the  number  of  pins  held 

~_      '  Q  by   the   rear   chainman    represents    the 

FIG.  3.    Survey  pins.       J     t          f  r  ,,  ,          i       <•        i 

number  of  full  lengths,  for  the  pins  used 

temporarily  for   the   short   lengths  have   been   exchanged 
each  time. 

In  commencing  to  measure  a  line  a  steel  or  iron  pin  is 
stuck  in  the  ground,  or,  if 
the  mark  is  plain,  the  pin 
is  held  by  the  rear  chain- 
man. When  a  tape  length 
is  measured  the  head  chain- 
man  sticks  a  pin  in  the 
ground  at  the  point  marked 
by  the  plumb-bob.  An- 
other length  is  then  meas- 
ured, the  rear  chainman 
pulling  up  the  pin  at  his 
end  after  the  head  chain-  ^ 
man  has  set  his  pin.  The 
number  of  pins  held  by  the 
rear  chainman  always  indicates  the  number  of  tape,  or 
chain,  lengths  measured,  the  pin  in  the  ground  not  being 
counted  for  it  merely  fixes  the  point  from  which  to  measure. 


FIG.  4. 


INTRODUCTORY  9 

The  head  chainman  starts  with  ten  pins,  the  total  number 
being  eleven.  When  ten  lengths  have  been  measured  the 
rear  chainman  goes  forward  and  hands  ten  pins  to  the  head 
chainman  and  they  check  by  counting  the  pins,  making  a 
record  of  the  "tally,"  so  that  the  correct  number  of  tape 
lengths  can  be  told  when  the  line  is  fully  measured.  The 
head  chainman  holds  the  zero  end  of  the  tape  and  thus  the 
number  of  feet,  or  links,  past  the  last  full  tape  length  may 
be  read  directly.  Pins  usually  have  a  length  of  about  one 
foot  and  are  stuck  in  the  ground  slanting  to  one  side  so  the 
plumb-bob  may  be  held  over  the  point  where  the  pin  enters 
the  ground.  In  order  to  find  the  pins  readily  it  is  customary 
to  tie  white,  or  brightly  colored,  strips  of  cloth  in  the  ring 
at  the  upper  end. 

Correct  measuring  cannot  be  done  when  weighted  pins 
are  used  instead  of  plumb-bobs,  although  weighted  pins 
were  formerly  used.  A  pin  cannot  be  accurately  dropped, 
no  matter  how  well  weighted. 

ROUTE   SURVEYS 

Route  surveys  for  roads,  ditches,  railways,  etc.,  are 
marked  by  stakes  driven  in  the  ground  at  each  tape  length, 
the  distances  between  stakes  100  ft.  apart  being  termed 
"stations,"  the  numbering  of  the  stations  proceeding  from 
zero.  Each  stake  being  numbered,  the 
number  on  any  stake  multiplied  by  100 
gives  the  distance  in  feet  from  Station  o, 
the  starting  point. 

In  land  surveys  only  the  distance 
between  corners  is  wanted  so  pins  are 
used  merely  as  counters  of  tape  lengths. 
In  route  surveys  the  elevation  of  the 
ground  at  each  station  point  is  required  _ 
for  the  purpose  of  fixing  grade  lines  and  FlG'  5nes^tbakaend  wlt' 
computing  quantities  of  earthwork,  there- 
fore the  stakes  are  numbered  and  left  in  the  ground  for 
the  use  of  the  leveler  and  slopeman.  These  stakes  are  set 
slanting  for  the  same  reason  that  pins  are  thus  set.  At 
stations  where  the  instrument  is  placed  for  the  purpose  of 
reading  angles  a  square  stake,  called  a  hub,  is  driven  flush 


10  PRACTICAL   SURVEYING 

with  the  ground  and  a  tack  driven  in  the  top  to  mark  the 
point  exactly.  A  stake  is  driven  to  one  side  as  a  witness 
and  on  the  witness  stake  the  necessary  identification  marks 
are  placed. 

After  the  stations  have  been  set,  by  the  transit  (or  com- 
pass) party,  elevations  of  the  ground  are  taken  at  each  stake 
and  recorded  in  a  book.  These  elevations  when  plotted  on 
ruled  paper  give  a  "profile"  of  the  route  and  on  the  profile 
is  fixed  the  grade  line.  The  slope  of  the  ground  to  the 
right  and  left  of  each  station  is  also  measured  and  from  the 
slope-notes  are  plotted  cross-sections  from  which  earth- 
work quantities  are  computed  after  a  grade  is  adopted. 

MEASURING  ON  SLOPES 

If  all  ground  sloped  the  same  degree  all  measurements 
might  be  made  on  the  surface  and  no  errors  would  arise. 
Since  some  ground  is  level  and  some  is  very  steep,  with 
many  degrees  of  slope  between,  it  is  necessary  to  adopt  a 
standard,  so  100  ft.  will  be  100  ft.,  no  matter  what  the 
nature  of  the  ground.  This  is  accomplished  by  holding 
the  tape  level  and  taut.  A  level  line  is  parallel  to  the  sur- 
face of  still  water  and  as  the  earth  is  practically  a  sphere 
a  level  line  is  really  the  circumference  of  a  great  circle.  The 
diameter  of  the  circle  is  so  great  and  the  distances,  com- 


FIG.  6.    McCullough  tape  level. 

paratively,  so  short  that  horizontal  lines  are  assumed  for 
all  practical  purposes  to  be  level,  a  horizontal  line  being 
perpendicular  to  a  vertical  line. 

Unskilled  men  experience  considerable  difficulty  in  hold- 
ing a  tape  level.  The  error  is  cumulative  and  the  sloping 
tape  measures  "  short"  each  time,  so  the  total  length  is  re- 
corded as  greater  than  it  is  actually  marked  on  the  ground. 
A  number  of  devices  are  used  to  assist  inexperienced  chain- 
men,  the  author,  in  1892,  patenting  a  small  level  for  the 


INTRODUCTORY 


II 


purpose.  The  chief  merit  of  this  device  is  that  it  may  be 
carried  in  the  vest  pocket  and  be  quickly  put  on  or  taken 
off  any  tape.  The  patent  having  expired  a  number  of 
instrument  dealers  now  advertise  similar  levels,  a  few 
crediting  the  inventor  by  name.  The  level  is  placed  on  the 
tape  about  one  foot  from  the  end  and  the  tape  is  pulled  tight. 
By  first  experimenting  with  the  tape  and  level  on  a  level 
surface,  as  a  floor,  the  positions  of  the  bubble  for  different 
tape  lengths  and  different  amounts  of  sag  are  noted.  With- 
out some  such  device  a  tape  may  be  held  horizontally  as 
follows:  While  the  rear  chainman  holds  his  end  stationary 
the  head  chainman  raises  and  lowers  his  end  until  he  ascer- 
tains the  longest  distance  which  can  be  obtained,  each  man 
using  a  plumb-bob.  On  route  surveys  it  is  usual  to  give 
each  man  a  hand  level,  the  proper  use  of  which  will  be  ex- 
plained later. 

The  numbering  of  stations  has  been  described.     Stakes 
set  between  are  marked  with  the  number  of  feet  from  the 


7 

Needle 

+50 

+30 

""*"    Stream 

6 

BlUft 

+59 

Uj 

<o. 

+50 

^ 

352 

5 

5: 

Jyv 

+67.3 

L  ZTIl'R 

N.4Z°52'E. 

2 

+  50 

4 

-    U' 

+50 

\  & 

3 

\  * 

75"&&x 

+50 

\    5s 

^j?& 

2 

j  30' 

Sg 

+50 

& 

1 

"SZ. 

+50 

q,  , 

-T 

0 

N.I5°4I'E. 

^S 

\^ 

_y 

FIG.  7.     Proper  methods  for  keeping  notes  of  line  survey. 

preceding  station,  preceded  by  a  +  sign.  Thus  4  +  50 
indicates  a  point  on  the  line  450  ft.  from  the  starting  point. 
It  is  read  "Station  4  plus  50."  The  transitman  enters  his 


12  PRACTICAL   SURVEYING 

notes  in  a  book  starting  on  the  bottom  line  on  the  left- 
hand  page,  the  notes  reading  from  the  bottom  up.  On  the 
right-hand  page  the  center  line  represents  the  line  of  survey 
and  sketches  may  be  made  with  objects  shown  in  the  proper 
relative  positions.  A  leveler  makes  no  sketches  so  stations 
are  numbered  from  the  top  down  in  his  field  book  in  order 
to  add  and  subtract  readily. 

Fig.  7  shows  some  notes  of  a  transit  survey.  At  Sta. 
o  a  hub  is  set  and  the  A  mark  placed  on  line  indicates  an 
instrument  point.  The  correct  bearing  is  set  down  in  the 
third  column  and  the  reading  given  by  the  needle  is  placed 
on  the  sketch  line  of  the  survey  as  a  check  on  the  "right" 
(R)  and  "left"  (L)  readings.  The  needle  is  always  read 
to  check  angle  readings  taken  on  the  horizontal  plate.  At 
Sta.  4  +  67.3  a  change  in  direction  occurred  and  this  was 
shown  on  the  sketch  line  in  order  to  give  the  draftsman  a 
check,  but  the  sketch  line  is  always  kept  straight.  Bearings 
are  obtained  from  preceding  angles  by  addition  or  subtrac- 
tion. 

When  very  accurate  work  is  required  all  measurements 
are  made  parallel  to  the  surface  of  the  ground,  between 
marks  in  the  tops  of  hubs  driven  at  each  tape  length. 
With  a  level  and  rod  the  elevations  of  the  tops  of  the  hubs 
are  taken.  Then 


H= 
in  which  expression 

H  =  horizontal  length, 
L  =  length  on  slope, 

D  =  difference  in  elevation  between  ends  of  tape  on 
slope. 

(Note.  —  In  all  problems  involving  squares  or  square 
roots  of  numbers  use  the  table  of  squares  in  Chapter  II. 
Information  on  the  solving  of  formulas  is  given  in  Ap- 
pendix A.) 

Problem.  —  A  line  was  measured  on  the  surface  of  the 
ground  with  a  tape  50  ft.  long,  hubs  being  set  at  each  tape 
length.  Elevations  were  taken  on  each  hub  and  the  differ- 
ences in  elevation  thus  obtained.  What  is  the  horizontal 
length  of  the  line? 


INTRODUCTORY 


Stations. 

Elevations 
above  base. 

Difference 
in  elevations. 

Stations. 

Elevations 
above  base. 

Difference 
in  elevations. 

O 

93-2 

0.0 

+  50 

109.  2 

0.2 

+50 

95-6 

2-4 

4 

II4-5 

5-3 

i 

/       98.2 

2.6 

+  50 

115.2 

°-7 

+  50 

101  .4 

3-2 

5 

115.8 

0.6 

2 

105.  1 

3-7 

+50 

117.0 

I.  2 

+50 

108.1 

3-o 

6 

119.4 

2-4 

3 

109.0 

0.9 

+50 

124-3 

4-9 

ANGULAR  MEASURING  ON  SLOPES 

In  rough  country  where  extreme  accuracy  is  not  re- 
quired and  speed  is  essential  it  is  a  common  practice  to 
measure  on  the  slope.  Sometimes  the  tapes  used  for  this 
work  are  several  hundred  feet  long.  They  are  held  at  a 
height  to  go  over  small  brush,  etc.,  and  plumb  bobs  at 
each  end  transfer  the  length  to  the  ground.  With  a  clinom- 
eter, or  transit,  the  vertical  angle  is  measured  from  the 
horizontal.  The  secant  of  an  angle  is  the  length  of  the 
hypothenuse  of  a  triangle  having  a  horizontal  base  =  I. 
Take  from  a  table  of  secants  (pp.  194—  200)  the  secant  of 
the  angle;  then 


in  which 


H  =  horizontal  distance. 
L  =  length  on  the  slope. 
A  —  angle  of  slope. 


The  following  table  shows  deductions  to  make  in  feet  for 
each  one  hundred  feet  measured  on  a  slope  provided  the 
surveyor  has  an  instrument  for  obtaining  the  angle  of 
slope.  Column  A  shows  the  angle  and  column  B  the 
number  of  feet  to  be  deducted. 


A 

B 

A 

B 

A 

B 

A 

B 

1° 

0.015 

6° 

0.548 

11° 

1.837 

16° 

3.874 

2 

0.061 

7 

0-745 

12 

2.185 

17 

4.37 

3 

0.137 

8 

0-973 

13 

2.563 

18 

4.894 

4 

0.244 

9 

1.231 

14 

2.970 

19 

5.448 

5 

0.381 

10 

i-5J9 

15 

3-407 

20 

6.031 

CHAPTER    II 
CHAIN   SURVEYING 

Steel  tapes  have  practically  supplanted  chains  for  meas- 
uring, but  the  word  chain  will  be  used  for  years  to  come 
and  the  men  who  make  linear  measurements  will  no  doubt 
always  be  called  chainmen.  The  term  "chain  surveying" 
applies  only  to  surveying  done  without  the  use  of  instru- 
ments to  measure  angles  or  read  bearings. 

The  surveyors'  chain  was  invented  by  Edward  Gunter  in 
1620  and  is  often  referred  to  as  Gunter's  chain.  It  is 
66  ft.  long  and  is  divided  into  100  links.  One  acre  contains 

1 60  square  rods  and  the  in- 
vention of  the  chain  by 
Gunter  simplified  area  calcu- 
lations. Ten  square  chains 
equal  one  square  acre,  so  the 
division  of  the  chain  into  100 
equal  parts  was  a  practical 
demonstration  of  the  value 
and  simplicity  of  the  decimal 
system  of  notation.  In  com- 
mon units  one  link  is  equal 
to  0.66  ft.  or  7.92  inches. 
The  surveyors'  chain  is  four  rods  long,  the  rod  being  an 
old  unit  for  land  measurement  in  Great  Britain.  An  old 
legend  describes  the  origin  of  the  rod  for  this  purpose  as 
due  to  the  fact  that  in  olden  times  farmers  carried  ox- 
goads,  or  rods,  which  were  conveniently  used  for  measur- 
ing land  and  a  royal  decree  fixed  a  standard  length.  In 
common  units  of  today  the  length  of  the  rod  is  sixteen  and 
one-half  feet,  this  being  the  rod  used  by  Gunter.  One 
mile  contains  eight  furlongs  and  a  furlong  has  a  length  of 
forty  rods.  Gunter  divided  the  furlong  into  ten  chains, 
which  thus  made  the  chain  four  rods  long. 

14 


FIG.  8.     Surveyors'  chain. 


CHAIN   SURVEYING  15 

As  long  as  the  acre  and  mile  remain  in  our  system  of 
measurements  the  Gunter  divisions  will  be  retained,  but 
as  time  passes  more  men  will  use  the  loo-ft.  tape  with  100 
links  each  one  foot  long.  Every  piece  of  land  in  the  United 
States  was  originally  surveyed  with  a  Gunter  chain,  so  to 
reduce  the  descriptions  to  modern  terms  involves  consider- 
able labor  and  already  has  given  rise  to  mistakes.  In  this 
work  the  word  "chain"  will  refer  to  the  surveyors',  or 
Gunter,  standard  with  a  link  of  7.92  ins.  as  the  unit.  The 
word  "tape"  will  refer  to  the  engineers'  standard,  a  tape 
100  ft.  long  with  the  decimally  divided  foot  as  a  unit. 

The  surveyors'  standard  for  land  measurement  made  in 
the  form  of  a  chain  is  heavy  and  the  sag  in  consequence  is 
so  great  that  each  length  is  less  than  it  should  be,  con- 
sidering a  horizontal  line  to  be  a  universal  standard.  When 
a  standard  is  short  it  is  applied  oftener  than  necessary 
when  measuring  a  line,  thus  the  recorded  length  is  too  great. 
Some  manufacturers  tried  to  overcome  the  effect  of  sag 
by  making  each  chain  a  trifle  long,  or  by  having  one  link 
at  each  end  attached  to  a  threaded  bar  by  means  of  which 
the  chain  could  be  lengthened  to  overcome  the  effect  of 
sag:  or  to  shorten  the  chain  when  the  rings  and  ends  of 
links  became  worn.  Some  chains  of  very  light  steel  have 
links  connected  end  to  end  but  this  arrangement  lacks  flex- 
ibility. It  is  usual  to  make  the  links  less  than  7  ins. 
long  and  connect  them  by  using  two  rings  between,  which 
provides  600  wearing  points.  When  a  chain  has  three  rings 
between  the  links  there  are  800  wearing  points.  A  slight 
pull  has  a  tendency  to  open  the  joints  and  in  the  better 
grade  of  chains  all  joints  are  closed  by  brazing. 

When  a  standard  is  long  it  is  not  applied  as  often  as  neces- 
sary when  measuring  a  line,  so  the  recorded  length  is 
short.  The  weight  of  a  chain  tending  to  make  it  short  and 
the  wearing  of  the  joints  tending  to  make  it  long  there 
never  was  a  definite  standard  of  length  in  the  days  when 
the  surveyors'  unit  of  measure  was  made  in  the  form  of  a 
chain.  Adjusting  screws  at  the  ends  only  affected  the 
whole  length  and  fractional  chain  lengths  were  never 
correct.  Links  frequently  became  bent  and  the  small  rings 
were  often  snarled,  thus  shortening  the  chain.  The  numer- 
ous joints  made  it  difficult  to  pull  a  chain  through  brush 


l6  PRACTICAL  SURVEYING 

and  places  filled  with  small  obstructions.  Fractional  parts 
of  links  had  to  be  estimated,  so  for  close  measurements 
surveyors  carried  rods  ten  links  long  with  the  links  divided 
decimally,  to  measure  the  ends  of  lines,  the  poles  being 
convenient  for  sighting  purposes.  In  cities  where  a  chain 


FIG.  9.     Contact  rod. 

would  be  out  of  place  it  was  customary  up  to  quite  recent 
times  to  measure  with  contact  rods.  These  were  made 
from  properly  seasoned  wood  and  were  shod  with  brass  at 
each  end.  The  length  was  a  matter  of  individual  prefer- 
ence, usually  ten  feet.  In  the  top  was  inserted  a  spirit 
level,  in  some  rods  a  level  being  placed  at  each  end.  Two 
or  more  rods  were  used,  being  placed  carefully  end  to  end. 
A  later  type  had  marks  near  the  ends  and  plumb-bobs  were 
used  with  one  rod,  instead  of  using  several  rods  placed  end 
to  end. 

Tapes  were  used  for  many  years  in  Europe  before  they 
were  adopted  in  the  United  States.     Since  1880  tapes  have 


(6)  (c) 

FIG.  10.  Steel  tapes:  (a)  Graduated  in  links,  (b)  Graduated  in  feet  and 
tenths,  (c)  Graduated  in  feet  and  hundredths.  Open  reels  protect 
against  rusting  of  tapes. 

so  grown  in  favor  that  the  old-style  chain  is  regarded  as  a 
curiosity.  Tapes  are  much  lighter  than  chains  so  the  sag 
is  less.  The  length  is  standard  at  a  temperature  of  practi- 
cally 60°  F.  with  the  tape  lying  on  a  level  surface  with  a  pull 
of  10  Ibs.  applied  at  the  ends.  The  coefficient  of  expan- 
sion for  steel  is  0.0000067  the  length  for  one  degree  F.  so 


CHAIN  SURVEYING  17 

that  a  tape  100  ft.  long  at  a  temperature  of  60°  F.  will 
be  99.981  ft.  long  at  a  temperature  of  32°  F.  and  will  be 
100.0134  ft.  long  at  80°  F.,  the  formula  being 

Lt  =Z,(id=Cr), 
in  which 

Lt  =  length  of  tape  at  assumed  temperature, 
L  =  length  of  tape  at  standard  temperature, 
C  =  coefficient  =  0.0000067, 

T  =  difference  in  degrees  between  standard  and  as- 
sumed temperature. 

When  a  tape  is  sold  the  maker  should  state  the  temper- 
ature and  pull  at  which  the  length  is  standard.  The 
United  States  Bureau  of  Standards,  Washington,  D.  C., 
will  test  and  certify  tapes  for  a  small  fee.  When  work  of  a 
very  important  character  is  to  be  performed  by  an  engineer 
or  surveyor  the  temperature  should  be  read  and  the  proper 
correction  applied  for  each  measurement.  The  effect  of 
temperature  is  to  shorten  a  tape  when  cold  and  lengthen  it 
when  warm.  Errors  continually  alter  in  amount  and  direc- 
tion during  the  day  but  they  practically  balance  in  ordi- 
nary work.  The  most  accurate  measurements  are  made 
during  cool  foggy  weather  or  at  night,  the  latter  being  the 
time  selected  by  Government  surveyors  for  the  highest- 
grade  work.  The  "Invar"  tape  is  made  of  an  alloy  of 
nickel  and  steel  possessing  a  very  small  coefficient  of  ex- 
pansion. It  is  used  on  work  of  the  highest  grade. 

The  effect  of  sag  is  to  shorten  measurements  and  being 
constant  is  important.  When  a  new  tape  is  purchased  it 
should  be  laid  on  a  level  floor  or  walk  at  a  temperature  as 
near  the  standard  as  possible.  A  spring  balance  should  be 
fastened  to  one  end  and  the  tape  pulled  until  the  standard 
pull  is  indicated,  the  ends  being  carefully  marked  on  the 
level  surface.  Two  men  should  now  hold  the  tape  at  a 
definite  height  above  the  level  surface  and,  holding  a  plumb- 
bob  at  each  end,  pull  the  tape  until  the  points  of  the  bobs 
strike  the  marks,  thus  eliminating  sag.  The  pull  then 
registered  on  the  spring  balance  will  be  from  50  to  100  per 
cent  greater  than  the  standard  pull  and  should  be  the  pull 
used  with  this  tape  thereafter.  Only  on  exceedingly  im- 
portant work  is  it  necessary  to  use  a  spring  balance,  a  chain- 


l8  PRACTICAL  SURVEYING 

man  after  a  few  tests  becoming  expert  in  applying  a  pull 
close  enough  for  ordinary  work.  Having  obtained  the 
proper  pull  as  described,  this  becomes  the  standard  pull 
for  this  particular  tape  in  order  to  eliminate  the  effect  of 
sag.  There  will  be  some  sag  with  any  less  pull  and  the  tape 
will  measure  short  of  the  proper  length.  By  using  ordinary 
care  no  correction  is  required  to  overcome  the  effect  of  sag 
due  to  lack  of  tension  but  a  wind  blowing  a  tape  to  one  side 
makes  it  curve,  or  deflect,  and  this  is  a  case  calling  for  a 
correction  on  important  work.  The  deflection  must  be 
obtained  and  the  correction  added  to  the  measured  length. 

Let  C  =  correction  for  deflection  or  sag  in  feet, 
d  =  amount  of  deflection  or  sag  in  feet, 
L  =  length  of  tape  in  feet, 


r 

then  C  =  —  = 

3^ 

When  a  pull  greater  than  the  standard  pull  is  used  the 
tape  will  stretch.  The  correction  must  be  subtracted  from 
the  measured  length,  on  important  work.  The  standard 
pull  here  referred  to  is  the  new  standard  obtained  by  meas- 
uring with  the  suspended  tape  above  a  level  surface. 

LetP  =  difference  in  pounds  between  the  standard  and 

actual  pull, 

A  =  area  of  cross-section  of  tape  in  square  inches, 
E  =  modulus  of  elasticity  of  steel  =  30,000,000, 
L  =  length  of  tape  in  feet, 
S  =  stretch  of  tape  in  feet, 


then 


ERRORS  IN  MEASUREMENT 


Errors  in  measurement  are  as  follows: 

i.  The  tape  may  not  be  held  in  a  horizontal  position. 
This  shortens  the  length  and  the  error  is  constant  and 
"cumulative."  A  cumulative  error  is  one  that  grows  by 
repetition  so  errors  of  this  sort  cannot  be  too  carefully 
guarded  against.  It  requires  long  practice  before  men 
learn  to  hold  a  tape  with  the  ends  at  the  same  elevation, 


CHAIN   SURVEYING  19 

so  work  done  with  inexperienced  chainmen  is  always  faulty. 
A  level  on  the  tape,  or  a  hand  level,  is  desirable  until  expe- 
rience is  gained. 

2.  The  tape  is  not  held  with  a  constant,  or  with  a  proper, 
tension.     This  causes  an  undue  amount  of  sag,  the  error 
being  cumulative. 

3.  Chainmen  are  not  carefully  kept  to  line  ("lined  in"). 
This  gives  a  measured  broken  line  instead  of  a  straight  line, 
the  error  being  cumulative. 

4.  The    plumb-bobs    may    be    too    light.     The    proper 
weight  for  a  plumb-bob  is  one  of  the  points  on  which  it 
is   easy  to  raise   discussion   between   practical   surveyors. 
Wind  delays  work  when  light  plumb-bobs  are  used  and  the 
error  is  more  likely  to  be  cumulative  than  compensating 
(that  is,  balancing).     A  light  plumb-bob  weighs  less  than 
one  pound.     The  writer  prefers  plumb-bobs  weighing  not 
less  than  one  and  one-half  pounds  with  a  decided  prefer- 
ence for  a  two-pound  bob. 

5.  Pins  may  be  placed  ahead,  or  back,  of  the  point.     If 
ordinary  care  is  used  this  error  will  be  compensating,  for 
with  'a  series  of  repetitions  the  number  of  pins  placed  ahead 
will  equal  the  number  placed  back  of  a  point,  the  law  of 
averages  applying  in  this  case.     The  error  is  only  important 
on  short  lines. 

6.  A  mistake  may  be  made  in  counting  pins,  thus  omit- 
ting a  chain  or  tape  length.     This  often  happens  and  can 
only  be  avoided  by  careful  attention  to  the  work,  each 
man  checking  the  count  of  the  other  at  the  ends  of  "  tallies.  " 

7.  The  chain  or  tape  may  not  be  of  standard  length. 
The  length  should  be  tested  on  a  level  surface  side  by  side 
with  a  standard  in  order  to  determine  a  proper  correction. 
The  measure  may  be  short  or  long. 

Let  M  =  length  of  line  as  measured  with  faulty  tape  or 

chain, 
T  =  true  length  provided  the  line  were  measured 

with  the  standard, 
/  =  assumed   length   of   tape  or  chain    (standard 

length), 
a  =  actual  length, 


then  r 


20  PRACTICAL   SURVEYI-NG 

Example.  —  A  line  was  measured  and  the  length  recorded 
as  1122.5  ft.  The  tape  was  later  discovered  to  be  99.8  ft. 
long  instead  of  100  ft.  What  is  the  true  length  of  the  line? 


CORRECTING  ERRONEOUS  AREAS 

A  tract  of  land  is  sometimes  surveyed  with  a  chain  or 
tape  not  of  standard  length  and  the  area  is  computed  before 
the  error  is  discovered.  It  is  not  necessary  to  re-compute 
the  area  by  first  correcting  each  line.  The  correct  area  may 
be  obtained  by  making  use  of  the  geometrical  proposition 
that  similar  polygons  are  to  each  other  as  the  squares  of 
the  like  sides. 

Let  A  =  true  area  of  field, 

C  =  computed  area  of  field, 
/  =  standard  length  of  chain  or  tape, 
a  =  actual  length  of  chain  or  tape, 


then 


PROBLEMS 


1.  A  line  was  recorded  as  having  a  length  of  43.80  chains 
but  the  chain  was  discovered  to  be  97  links  long.     What 
was  the  true  length  of  the  line? 

2.  A  tape  having  0.15  ft.  broken  off  one  end  was  used 
to  measure  a  line  which  was  reported  as  a  result  of  the 
measurement   to   be    1345.2  ft.  long.     Assuming  that  the 
broken  tape  was  used  as  if  it  were  50  ft.  long  what  was 
the  actual  length  of  the  line? 

3.  A  tape  standard  at  70°  F.  with  a  pull  of  18  Ibs.  when 
freely  suspended  was  used  to  measure  a  line  2910  ft.  long. 
What  length  was  returned  by  the  surveyor  who  used  the 
same  tape  with  a  pull  of  26  Ibs.  at  a  temperature  of  96°  F. 
and  made  no  corrections? 

4.  A  tape  200  ft.  long  was  used  to  measure  a  line  7800  ft. 
long  on  a  windy  day  with  a  deflection  of  0.73  ft.  for  each 
tape  length.     What  was  the  length  of  the  line  as  returned 


CHAIN  SURVEYING 


21 


by  the  surveyor  who  made  no  allowance  for  sag  caused 
by  wind? 

5.  A  line  was   measured  with   a    ipo-ft.   tape  and  the 
length  returned  as  6700  ft.,  a  stake  being  set  at  each  tape 
length.     The  lining-in  was  done  by  the  chainmen.     After- 
wards an  instrument  was  used  to  project  a  perfectly  straight 
line,  when  half  the  stakes  were  found  to  be  6  ins.  on  one 
side  and  half  were  found  to  be  7  ins.  on  the  opposite  side 
of  the  line.     What  was  the  actual  length? 

6.  The  area  of  a  field  was  returned  as  48.9  acres.     The 
tape  was  later  found  to  be  half  a  link  too  long.     What  was 
the  true  area? 

7.  A  chain  98.5  links  long  was  used  to  measure  a  field, 
the  area  of  which  was  known  to  be 

93.2  acres.     What  was  the  area  as 
found  with  the  defective  chain? 

8.  A  field  known  to  contain  83.4 
acres  was  measured  with  a  defec- 
tive chain  and  the  area  given  as 
being  80.7  acres.      What  was  the 
actual  length  of  the  chain? 

RANGING  LINES 

Surveyors  use  poles  of  wood  sev- 
eral feet  long  for  ranging  lines  and 
for  sights.      These  poles  are  shod 
with  a  metal  point  on  one  end  and 
are  painted  with  one   foot  spaces 
alternately  red  and  white.  The  top 
diameter  is  usually  one  inch  and 
the  diameter  at  the  bottom  is  one       a 
and  one-half  inches.     For  use  with  FIG.  n. 
instruments  having  telescopes  the 
sighting    poles    are   of    steel    five- 
eighths  of  an  inch  in  diameter.     Steel  poles  are  called  line 
rods.     For  very  accurate  work  line  rods  have  a  relatively 
short  point  at  one  end  for  setting  hubs,  the  other  end  having 
a  long  point  used  in  setting  tack  points  on  hubs. 

To  range  a  line  with  poles.  —  Three  or  more  points  are 
required.     One  man  can  do  the  work  but  time  is  saved 


Sighting  poles  and 
line  rods. 


22  PRACTICAL   SURVEYING 

when  two  or  more  men  are  used.  The  general  direction  of 
the  line  being  chosen  a  pole  is  set  on  line  at  any  distance 
from  the  starting  point  and  a  second  pole  is  set  ahead  in  line 
with  the  first  pole  and  the  starting  point.  The  first  pole 
is  then  carried  ahead  and  set  in  line  with  the  second  pole 
and  the  starting  point,  this  operation  being  repeated  until 
the  starting  point  cannot  be  plainly  seen.  A  third  pole  is 
then  used  and  the  line  carried  forward  by  moving  one  pole 
ahead  and  setting  it  in  line  with  the  two  standing  poles. 
A  small  stake  is  driven  in  each  hole  as  the  poles  are  lifted. 
It  is  better  to  measure  the  line  and  set  the  stakes  at  regular 
intervals.  In  setting  the  poles  a  plumb  line  should  be  used 
to  insure  verticality. 

Assume  that  a  line  has  been  ranged  with  poles  with  the 
intention  that  it  shall  strike  a  certain  point  and  it  fails  to 
do  so.  The  offset  distance  is  measured  from  the  point  to 
the  line  and  each  stake  must  be  moved  proportionately. 
In  Fig.  12  the  random  line  from  A  to  C  was  ranged  with 


FIG.  12.    Illustrating  random  line. 

poles,  the  intention  being  to  strike  the  point  B  which  was 
missed  by  60  links,  the  total  length  being  5  chains  and  75 
links.  At  the  end  of  each  chain  a  stake  was  set  and  these 
stakes  must  be  moved  over  to  the  true  line.  What  correc- 
tion is  to  be  made  at  each  stake? 

Solution.  —        -  =  10.435  links  per  chain.     The   stakes 

will  be  moved  10.43  links  at  I ;  20.87  nnks  at  2;  31.31  links 
at  3;  41.74  links  at  4  and  52.175  links  at  5,  stake  C  being 
discarded. 

To  pass  obstacles.  —  Measure  to  one  side  of  the  first  pole 
a  distance  sufficient  to  enable  a  line  to  pass  the  obstacle. 
Measure  an  equal  distance  from  the  third  pole  and  set  a 
middle  pole  between  these  two  by  sighting.  There  will  be 
three  poles  set  in  a  straight  line  parallel  to  the  line  first 


CHAIN  SURVEYING  23 

started.  This  second  line  is  ranged  forward  until  the  ob- 
stacle is  passed  when  the  original  line  is  resumed  by  meas- 
uring back  to  the  line  from  two  poles  set  in  line  on  the 
offset  line  beyond  the  obstacle. 

To  range  a  line  over  a  hill.  —  Let  A  and  J3,  Fig.  13,  be 
on  the  line.  One  man  stands  at  C,  where  he  can  see  both 
A  and  B.  A  second  man  is  then  lined  in  to  d,  between  c 


*':L 


FIG.  13.     Ranging  a  line  over  a  hill. 

and  A.  Standing  at  d  and  facing  B,  the  second  man  lines 
the  first  man  in  to  e,  on  line  between  d  and  B.  From  e 
the  first  man  then  lines  the  second  man  to/,  on  line  between 
e  and  A.  In  this  manner  the  poles  are  successively  brought 
closer  to  line  at  each  operation  until  finally  one  is  set  at  j 
and  the  other  at  k  on  the  line  from  A  to  B. 

Further  use  of  poles  and  tapes  calls  for  the  exercise  of 
considerable  ingenuity  and  the  problems  should  be  first 
worked  out  on  a  drawing  board.  When  the  principles  are 
mastered  the  work  may  be  repeated  on  a  larger  scale  in 
the  field. 


FIG.  14.    T  square. 
PRACTICAL   GEOMETRY 

The  student  should  have  the  following  drafting  tools  : 

1.  A  drawing  board  of  soft  pine,  f  in.  thick,  preferably 
1 8  ins.  by  24  ins.  in  size.     All  edges  should  be  straight  and 
true,  the  lower  left-hand  corner  being  a  perfect  right  angle. 

2.  AT  square  with  a  blade  24  ins.  long.     The  best  have 
transparent  celluloid  edges  and  the  heads  are  fixed.     A  T 


'PRACTICAL   SURVEYING 


square  has  a  head  to  slide  along  the  edge  of  a  drafting 
board  so  lines  ruled  along  the  edge  of  the  blade  as  the  head 
is  moved  will  be  parallel.  The  head  should  form  a  true 
right  angle  with  the  blade  and  if  the  edges  of  the  board  are 
true  and  two  of  the  edges  form  a  perfect  right  angle,  lines 
drawn  along  the  edge  of  the  T  square  will  form  right  angles 
when  the  head  is  placed  first  against  one  side  and  then 
against  a  side  perpendicular  to  the  first. 

For  architectural  and  mechanical 
drafting  and  for  map  drafting  where 
all  angles  are  right  angles,  or  are 
readily  drawn  from  a  horizontal  or 
vertical  line  as  a  base,  a  T  square 
is  useful  and  well-nigh  indispensa- 
ble. For  the  surveyor  a  straight- 
edge is  better  than  a  T  square.  All 

lines  do  not  so  run  that  right  angles 
15.       urattsmens'    tri- are     formed     and     a     heavy     steel 

straight-edge  may  be  laid  anywhere 
on  a  drafting  board  in  any  direction  and  it  will  not  be  dis- 
turbed when  triangles  are  slid  along  the  edge,  provided 
proper  care  is  exercised. 

3.  A  3O-deg.  triangle,  8  ins.  long. 

4.  A  45-deg.  triangle,  8  ins.  long. 

The  three  interior  angles  of  a  triangle  equal  two  right 
angles.     A  3O-deg.  triangle  contains  one  right  angle,  one 


30'x  60*90'  45x45x9i 

FIG  .    i  ^ .       D  raf  tsmens ' 
angles. 


FIG.  16.    Engineers'  triangular  scale. 

angle  of  30  deg.  and  one  angle  of  60  deg.  A  45-deg.  trian- 
gle contains  one  right  angle  (90  deg.)  and  two  45-deg. 
angles.  Transparent  celluloid  triangles  are  better  than 
triangles  of  rubber  or  of  wood. 

5.  An  engineers'  triangular  scale.  The  six  edges  are 
divided  into  one-inch  spaces  with  each  edge  divided  deci- 
mally for  convenience  in  plotting  lines.  The  usual  gradua- 


CHAIN  SURVEYING 


of  an  inch,  enabling  the 
scales  of   10,    100,    1000, 


tions  are  TV,  ?<i»  ^  ?V>  -fa,  eV 
draftsman   to   make  maps  with 
20,  200,  2000,  30,  300,  3000,  etc.,  feet  per 
inch,  inches,  feet  or  yards  per  mile,  etc. 

6.  A  3-H  lead  pencil,  although  it  is  well 
to  have  a  4-H  pencil  also.     The  4-H  pencil 
will  be  used  to  mark  points  and  draw  lines 
lightly,  the  3-H  pencil  being  used  to  draw 
more  prominent  lines. 

7.  Pencil  eraser. 

8.  Drawing  pen,  5  ins.  long. 

9.  Compass  with  pen  and  pencil  points, 
for  drawing  circles.     The  length  should  be 
6  ins.  or  8  ins. 

Other  tools  will  be  required  when  com- 
pass surveying  is  reached  but  for  present 
use  the  foregoing  will  be  "sufficient.  Use  a 
good  quality  of  drawing  paper,  the  most  FlG-  I7-  («)  Rut 
restful  color  to  the  eyes  being  a  light  brown 
or  buff.  For  holding  the  paper  to  the  draw- 
ing board  use  thumb  tacks  of  punched 
steel  f-in.  diameter. 

To  draw  parallel  lines.  —  Place  a  triangle  on  the  paper 
and  hold  it  by  pressing  with  the  tips  of  the  fingers  of  the 


(a) 


pass  with  pen 
point  and  length- 
ening bar. 


FIG.  i 8. 


FIG.  19. 


left   hand.     With    the   right   hand   slide   another   triangle 
along  one  side  of  the  first. 

In  Figs.  1 8  and  19  the  line  a  ...  b  having  been  drawn, 
the  triangles  are  held  so  the  edge  c  .  .  .  d  touches  the 
line  a  ...  b.  Holding  the  larger  triangle  firmly  the 


26 


PRACTICAL   SURVEYING 


smaller  one  is  held  closely  in  contact  with  it  and  moved  to 
the  point  c,  when  a  pencil  drawn  along  the  edge  will  describe 
the  line  c  .  .  .  d  parallel  with  a  .  .  .  b.  Instead  of  two 
triangles  a  straight-edge  or  T  square  may  be  used  with  one 
triangle.  By  using  two  triangles  in  combination  with  a 
straight-edge  or  T  square  a  number  of  angles  may  be 
formed  as  shown  in  Fig.  20. 


FIG.  20. 

The  surveyor  seldom  uses  triangles  for  setting  off  angles 
in  the  manner  shown  in  Fig.  20.  The  principal  use  to 
which  he  puts  these  useful  tools  is  to  transfer  lines  from  one 
part  of  a  map  to  another. 

PROBLEMS 

I .  To  erect  a  perpendicular  to  a  given  line  passing  through 
a  given  point  outside  the  line.  —  Let  B-C  be  the  line  and  A 
be  the  point. 

1st  method.  —  Draw  the  line  B—C.  Set  the  needle  point 
of  the  compasses  at  A  and  describe  the  arc  intercepting 
B-C  at  D  and  E.  Bisect  D-E  and  from  the  midway  point 
F  draw  the  line  F-A ,  which  will  be  the  required  perpendic- 
ular. Fig.  21. 

2nd  method.  —  Describe  arc  D-E  as  before.  With  needle 
point  at  E  set  pencil  point  at  A  and  describe  arc  A-E. 


CHAIN  SURVEYING  27 

With  needle  point  at  D  and  pencil  point  at  A  describe  arc 
A-F  intersecting  at  F  the  first  arc  drawn  from  E  as  the 
center.  The  line  A-F  will  be  the  required  perpendicular. 
Fig.  22. 

$rd  method.  —  Let   F  be   the  point.     From  a  point  A 
describe  an  arc  D-E-F  and  a  line  connecting  E  and  F  will 


/ 


V 


¥-' 


FIG.  21. 


FIG.  22. 


FIG.  23. 


be  the  required  perpendicular  provided  the  three  points  Z>, 
A  and  F  are  in  a  straight  line.  To  accomplish  this  draw 
a  line  from  F  to  the  line  jB-C  and  mark  D.  Set  off  the 
middle  point  A,  so  that  DA  =  AF.  Fig.  23. 

From  the  foregoing  problems  a  perpendicular  is  seen  to 
be  a  line  intersecting  another  line  in  such  a  manner  that 
the  angle  formed  on  one  side  of  the  line  is  equal  to  the  angle 
on  the  other  side.  By  the  second  method  four  equal  angles 
are  formed,  each  being  a  right  angle. 
A  perpendicular  line  is  also  called  a 
normal  line. 

With    the   needle   point   set    at   the 
intersection  of  two  lines  normal  to  each 
other  describe  with  the  pencil  point  a 
complete  circle.     Fig.  24.     Then 
<AOB=  <BOC=  <COD=  <DOA, 

the  sign/  standing  for  the  word  "angle." 

Circles  are  divided  into  360  equal  divisions  called  degrees 
and  one-fourth  of  360  =  90,  so  each  of  the  equal  angles 
formed  by  the  intersection  of  perpendicular  lines,  or  lines 
normal  to  each  other,  contains  90  deg.  Such  an  angle 
is  called  a  right  angle,  and  contains  a  "quadrant,"  or 
quarter  of  a  circle. 

An  angle  is  the  amount  of  divergence  of  two  intersecting 


28  PRACTICAL   SURVEYING 

lines.  The  amount  of  divergence,  that  is,  the  difference  in 
direction  between  the  line  A-0  and  the  line  B-O,  is  90  deg. 
(90°).  Between  A-0  and  C-O,  proceeding  to  the  right  in 
a  clockwise  direction,  the  amount  of  divergence  is  180  deg., 
the  angle  being  known  as  a  straight  angle.  Between  A-0 
and  D-0  the  amount  of  divergence  in  a  clockwise  direc- 
tion =  270  deg.  and  in  an  anticlockwise  direction  =  90 
deg.  The  three  figures  show  the  line  A-D  to  be  equal  to 
the  line  A-E.  This  is  proof  that  the  angles  formed  by  the 
meeting  of  the  perpendicular  lines  are  equal,  for  lines  op- 
posite angles  are  proportional  to  the  angles. 

When  the  problem  has  been  worked  on  paper  the 
operations  should  be  repeated  on  the  ground.  The  lengths 
of  the  lines  are  unimportant  except  as  they  govern  the 
accuracy  of  the  work;  longer  lines  producing  more  accu- 
rate results.  The  first  method  is  generally  used  when 
the  line  B-C  is  along  a  wall  arid  all  the  work  must  be 
performed  on  one  side.  Drive  a  stake  or  pin  at  A  and  hold 
the  end  of  the  tape  there.  Take  any  distance  A-D  to  the 
wall  and  make  a  mark  at  D.  On  the  other  side  measure 
A-E  =  A-D  and  mark  the  point  E.  Halfway  between  D 
and  E  make  the  mark  F,  which  will 
be  the  point  where  a  perpendicular 
through  A  will  meet  the  wall. 

Sometimes  it  may  be  necessary  to 
erect  a  perpendicular  from  a  point. 
Let  the  point  be  F  (Fig.  25).  From 


FIG.  25.  F  measure  F-D  and  F-E,  equal  to 

each  other.  With  any  radius  describe 

an  arc  with  D  as  a  center  and  with  the  same  radius 
describe  an  arc  with  E  as  a  center.  The  arcs  will  intersect 
at  A. 

The  second  method  may  be  used  when  no  wall  or  other 
obstruction  prevents  a  measurement  on  both  sides  of  line 
B-C.  For  the  reason  that  long  radii  may  be  used  and  the 
required  point  is  midway  between  two  accurately  deter- 
mined points,  this  method  is  apt  to  be  more  accurate  than 
the  first  method. 

The  third  method  is  not  so  good  as  either  of  the  two  others, 
but  is  often  used  when  a  perpendicular  is  to  be  set  off  from 
a  point  on  a  line  and  not  through  a  point  off  the  line. 


CHAIN  SURVEYING 


Let  E  (Fig.  26)  be  a  point  on  the  line  B-C,  usually  the 
end  of  the  line,  in  which  case  the  line  is  B-E.  Select  some 
point  A  as  a  center  and  with  a  radius  =  A-E  describe  the 
arc  D-E-F  intersecting  the  line  at  D  and  having  F,  as 
nearly  as  the  eye  can  judge,  on  the  line  D-A  prolonged. 
With  £  as  a  center  with  radius  =  E-D  describe  an  arc 
intersecting  the  first  arc  at  F.  A  line  through  E-F  will  be 
the  required  perpendicular  through  E. 


FIG.  26. 


FIG.  27.    Marking  inter- 
section of  two  lines. 


2.  To  mark  the  intersection  of  two  lines  in  the  field.  —  In 
Fig.  27  let  B-C  be  one  line.     It  is  assumed  that  the  second 
line  D-E  will  cross  B-C  near  the  point  0.     A  pole  may  be 
set  at  C  and  the  observer  standing  on  line  at  B  sights 
towards  C  over  pegs  driven  on  line  at  F  and  G,  with  small 
nails  or  tacks  in  the  top  to  mark  the  exact  line.     Standing 
at  D  with  a  pole  at  E,  similar  pegs  are  set  at  H  and  7.     Tie 
strings  to  the  projecting  nails,  connecting  F  to  G  and  H 
to  /,   drawing  the  strings  taut.     Under  the  intersection 
of  the  strings  at  0  drive  a  peg  and  drive  a  small  nail  or 
tack  in  the  top  of  the  peg  to  mark  the  exact  intersection. 

If  the  line  D-E  is  an  arc,  after  setting  the  points  F  and 
G,  connect  them  with  a  string  and  describe  the  arc,  setting 
peg  with  tack  at  the  intersection.  In  this  case  pegs  H 
and  /  are  not  necessary.  This  latter  method  may  also  be 
used  when  line  D-E  is  straight,  the  peg  and  tack  at  0  being 
set  by  sighting  an  intersecting  line  after  the  string  is  tied 
to  F  and  G.  The  best  method  to  use  depends  upon 
circumstances,  governed  by  the  judgment  of  the  surveyor. 

3.  To  erect  from  a  point  on  a  given  straight  line  a  perpen- 
dicular to  the  line. 


3° 


PRACTICAL   SURVEYING 


The  square  on  the  hypothenuse  of  a  right-angled  triangle 
is  equal  to  the  sum  of  the  squares  upon  the  other  two  sides ; 
.  that  is 

Hyp2  =  base2  -f-  altitude2, 
or         Hyp.    =  \/base2  -f  altitude2. 

Designating  the  angles  by  capital 
letters  and  the  sides  opposite  by  small 
letters 


base 

FIG.  28. 


c2  =  a2  +  b2. 
a  = 


-  b2     and     b  =  Vc2  -  a2. 


Assume  values  for  some  triangle  as  follows: 

I 
then  < 


3;     b  =  4;     c  =  5; 

-  42  =A/25  -  16  =A/9  =  3, 


which  proves  that  when  the  sides  of  a  triangle  are  to  each 
other  as  3,  4  and  5,  the  angle  at  C  will  be  a  right  angle. 

Using  a  multiple  of  4,  set  off  the  base  A-C.  Using  the 
same  multiple  of  5,  describe  an  arc  from  A  as  a  center,  pass- 
ing through  B.  Using  the  same  multiple  of  3,  describe  an 
arc  from  C  as  a  center,  intersecting  the  first  arc  at  B.  A 
line  connecting  B  and  C  will  be  normal  (perpendicular)  to 
A-C. 

4.    To  set  out  on  the  ground  a  line  parallel  to  another  line.  — 
In  Fig.  29  the  line  C-D  is  to  be  laid  off  parallel  to  the  line 
A-B.      Three   methods  are    in    common 
use. 

ist  method.  —  With  radius  =  A-C  de- 
scribe an  arc  from  A  and  also  from  B.  ~ 
Sight  across  the  two  arcs  on  the  line  C-D  . 
and  set  pegs.  The  second  method  is  pref-  / 
erable  for  it  is  difficult  to  obtain  a  good  A 
intersection  on  flat  curves. 

2nd  method.  —  From  A  with  a  radius 
A-A'  describe  an  arc  intended  to  pass  through  C.  With 
the  same  radius  from  A'  describe  an  arc  intersecting  the 
first  arc  at  C  and  drive  a  peg.  Repeat  these  operations  at 
B  and  Bf,  setting  a  peg  at  D. 


fa 


6'    B 


FIG.  29. 


CHAIN  SURVEYING  31 

Whether  to  use  the  first  or  second  method  depends  upon 
circumstances  and  the  judgment  of  the  surveyor,  the 
accuracy  desired  being  a  governing  factor.  The  distance 
A-B  should  be  not  less  than  ten  times  the  distance  A-C, 
the  degree  of  accuracy  depending  upon  the  ratio  adopted 
and  the  care  with  which  the  work  is  done. 

yd  method.  —  This  method  is  based  on  the  geometrical 
truth  that  when  a  straight  line  crosses  two  parallel  lines 
the  alternate  angles  ABC  and  DCS  are  equal.  At  B  and  C, 
Fig.  30,  set  pegs  and  stretch  a  string  to  define  the  line  B-C. 
From  B  as  a  center  describe  the  arc  e-f  and  measure  the 


FIG.  30. 


FIG.  31.     Equilateral  triangle. 


chord  (the  straight  line  e-f).  From  C  with  the  same  radius 
describe  the  arc  g-h  and  set  off  the  chord  g-h  =  chord 
e-f.  From  C  a  straight  line  through  g  will  fix  the  location 
of  C—D.  This  problem  is  most  conveniently 
worked  when  an  equilateral  triangle  is  formed 
by  the  lines  A-B,  A-C  and  B-C. 

5.  To  describe  an  equilateral  triangle.  — 
In   Fig.    31    from  C  with   a   radius  =  A-C 
describe  an  arc  through  B.     From  A  with 
the  same  radius  describe  an  arc  intersecting 
the   first  arc  at  B.     Draw  the  lines  A-B 
and  A-C,  forming  the   triangle,  which  has 
three  equal  sides  and  three  equal  angles. 

(Note.  — An   isosceles    triangle    has    two  FlG         Isosceies 
equal    sides    and    two    equal    angles,    Fig.        triangle 
32.) 

6.  To  pass  an  obstacle  on  line.  —  When  an  obstacle  is 
encountered  in  ranging  a  line  it  may  be  passed  by  off- 
setting as  already  described.     A  common  method  is  shown 


32 


PRACTICAL   SURVEYING 


Line     D 


{-Backsight 


'oresight 


in  Fig.  33.     From  A  and  B  perpendiculars  of  equal  length 
are  measured  and  points  A'  and  B'  set.     Sighting  from  A' 

past  B'  stakes  are  set  at 
C'  and  D'.  Perpendiculars 
from  the  offset  line  are 
measured  from  C'  to  C  and 
from  D'  to  D,  equal  in 
FIG.  33.  length  to  A  A'  and  BB', 

then  A ,  B,  C,  D  are  on  the 

same  line.  Another  method  not  often  used  is  shown  in 
Fig.  34.  On  the  line  A-B  set  the  points  I  and  2.  With  a 
base  equal  to  the  distance  between 
these  points  erect  the  equilateral 
triangle  1-2-3.  Produce  the  side 
1-3  to  a  point  5  opposite  the  ob- 
stacle and  construct  the  equilateral 
triangle  4-5-6  with  sides  equal  to 
1-2-3.  Produce  the  line  5-8  with 
5-6  =  4-5  and  6-7  =  3-4.  Make 
7-8  =  5-6  and  construct  the  equi- 
lateral triangle  7-8-9.  The  base 
8-9  should  be  on  the  line  A-B. 


FIG.  34. 


The  method  just  described  is  clumsy  and  the  base  8,  9 
is  very  short  so  a  slight  error  will  throw  the  line  off  con- 
siderably. A  parallel  line  set  off  by  means  of  carefully 
measured  perpendiculars  is  recommended  as  the  best  means 
for  passing  obstacles.  The  triangle  method  is  used  only 
when  the  survey  is  small  and  it  is  impossible  to  obtain  long 
sights  because  of  numerous  obstacles. 

7.  To  divide  a  line  into  any  number  of  equal 
parts.  — 

1st  method.  —  Measure  the  line  carefully 
and  divide  the  length  by  the  number  of  spaces 
into  which  it  will  be  divided.  This  is  the  field 
method. 

2nd  method.  —  This  is  an  office  method  and 
is  shown  in  Fig.  35.  It  is  useful  in  making 
scales  for  drafting.  Assume  that  on  a  map 
a  line  A-B  is  known  to  be  a  certain  length  but  the  paper 
has  shrunk  through  age  and  the  scale  is  not  known.  From 
one  end  lay  off  a  line  A-C  of  any  length  and  divide  it 


FIG.  35. 


CHAIN   SURVEYING  33 

into  the  number  of  parts,  by  scaling,  into  which  it  is  de- 
sired to  divide  A-B.  Draw  the  line  C-B  and  by  using  a 
straight-edge  and  triangle  draw  parallel  lines  through  A-B 
from  the  points  marked  on  A— C. 

PROBLEM 

A  map  is  to  be  made  to  a  scale  of  one-half  mile  to  an  inch 
and  no  scale  is  available  for  the  purpose.  One-half  mile 
contains  2640  ft.  so  on  a  piece  of  paper  draw  a  line  A-B 
one  inch  long  and  the  line  A—C  2.64  ins.  long,  using  the 
ipo  per  inch  graduation  of  the  triangular  scale.  On  the 
line  A-C  a  division  of  TV  in.  =  100  ft.,  so  through  each 
TVin.  graduation  on  A-C  draw 
a  line  through  A— B  parallel  with 
C-B.  Each  space  on  A-B  equals 
100  ft.,  except  the  last  space  which 
equals  40  ft. 

8.  To  divide  a  straight  line  in 
any  proportion,  as  3,  5,  7,  etc.  — 
Let  the  line  be  A-B  in  Fig.  36. 

From  A  draw  the  line  A-C,  making  A-E  =  3  and  E-C  =  5: 
From  B  draw  the  line  B-D  parallel  to  A-C,  making  B-F  = 
7  and  F-D  =  5.  Join  C  to  Fand  E  to  D,  thus  making  A-H 
=  3,  H-K  =  5  and  K-B  =  7. 

A  number  of  problems  might  be  given  similar  to  those 
already  treated  but  it  is  assumed  the  student  has  a  scale 
with  which  to  work  in  the  office  and  a  tape  with  which  to 
work  in  the  field,  so  some  problems  may  be  more  readily 
solved  by  arithmetical  methods  than  by  strictly  geometrical 
methods.  By  measurement  the  last  problem  would  be 
solved  as  follows:  Add  3  +  5  +  7  =  15.  Measure  the 
line  and  divide  by  15.  Then  A-H  =  A  of  A-B;  H-K  = 
T5s  of  A-B;  K-B  =  TV  of  A-B.  Annoying  differences  may 
creep  into  arithmetical  and  measuring  work  whereas  the 
geometrical  method  is  exact  if  proper  intersections  can  be 
obtained.  The  geometrical  method  may  be  used  in  either 
the  field  or  the  office  when  no  tapes,  scales  or  other  measur- 
ing instruments  are  available. 

9.  To  bisect  a  given  angle  ABC.  —  From  B  in  Fig.  37 
with  any  radius  cut  the  sides  in  A  and  C.     From  A  with  any 


34 


PRACTICAL   SURVEYING 


radius  A-D,  and  from  C  with  the  same  radius,  describe  arcs 

intersecting  at  D.     Join  B-D  and  then  /.ABD  =  /.CBD. 

10.    To  divide  a  right  angle  into  three  equal  parts.  —  From 

A  in  Fig.  38  with  radius  A-B  describe  the  arc  B-E-D-C. 


FIG.  37. 


FIG.  38. 


From  B  with  the  same  radius  cut  this  arc  in  D  and  from  C 
with  the  same  radius  cut  the  arc  in  E.  Then  Z  CAD  = 
ZZX4E  =  ZE.45  =  30  deg. 

(Note.  —  -  No  direct  method  is  known  for  closer  division 
except  continual  bisection,  or  dividing  by  trial.  With  a 
protractor,  an  instrument  to  be  later  explained,  any  re- 
quired angle  may  be  set  off.) 

ii.    To  construct  a  triangle  when  the  three  sides  are  known. 

-  In  Fig.  39  lay  off  the  line  A-C  to  scale.     From  C  with 

a  radius  =  C-B,  and  from  A  with  a  radius  =  A-B,  describe 


FIG.  39. 


FIG.  40. 


arcs  intersecting  at  B.  Then  A-B-C  is  the  required  trian- 
gle. 

12.  To  measure  an  inaccessible  distance,  for  example  a 
line  crossing  a  stream  or  swamp.  — 

1st  method.  —  The  principle  of  equal  triangles  illus- 
trated in  Fig.  40  is  used.  Set  out  a  triangle  A-B-C 


CHAIN   SURVEYING 


35 


with  right  angle  at  B.  Make  C-D  =  B-C  and  set  a  pole 
vertically  at  C.  Opposite  D  set  pole  at  E  on  a  line  normal 
to  the  line  B-D.  One  observer  stands  at  A  and  sighting 
past  C  directs  an  assistant  with  a  pole  into  position  at  F. 
Another  observer  stands  at  D  and  sighting  past  E  also 
directs  the  assistant  at  F.  The  lines  A-C  and  D-E  inter- 
sect at  F  and  D-F  =  A-B.  By  measuring  the  length  of 
A-B  the  distance  between  D  and  F  is  found,  the  two  trian- 
gles being  equal. 

2nd  method.  —  By  similar  triangles.      The  line  B-C  may 

be  made  equal  to  -  -  when  D-F  will  be  twice  the 

length  of  A-B.     If  B-C  =  lme  Z)~Cthen  line  D-F  will  be 

o 

three  times  the  length  of  A—B.  The  divisor  may  be  any 
number  instead  of  2  or  3.  Let  B-C  divided  by  D-C  be  any 
ratio,  then 

A-B  X  D-C 


D-F 


B-C 


In  Fig.  41  is  illustrated  another  form  in  which  triangles 
may  appear.     The  line  B-D  is  produced  across  the  river 
and  at  B  a  perpendicular  line,  B-C,  is  set 
off.     At    C  set   off  a  line  perpendicular  to  ° 


C-D,  intersecting  B-D  at  A  . 
then 


B-D 


Measure  A-B, 


A-B 


FIG.  41. 


The   length   A-B   is    proportional    to   B-D 

and  any  error  in   measurement  will   be  in 

the    same    proportion.      Line    B-C   should 

therefore  be  not  less  than  one-half  or  two-thirds  the  length 

of  B-D. 

13.  To  erect  a  perpendicular  to  a  line  from  an  inaccessible 
point.  —  In  Fig.  42  point  C  is  a  fence  post  from  which  a  line 
is  to  cross  the  river  to  an  intersection  with  the  fence  A—B, 
forming  an  angle  of  90  deg.  with  the  fence.  At  any  con- 
venient point,  H,  on  the  line  A-B  erect  perpendicular  lines 
on  both  sides.  By  intersection  fix  F  on  the  line  A-C  and 


PRACTICAL  SURVEYING 


make  H-F  =  G-H.  Sight  from  G  to  C  and  by  intersec- 
tion fix  /  on  the  line  A-B.  Sighting  from  F  past  /  and 
from  A  past  G,  fix  E  by  intersection.  From  E  sight  to  C 


7 


FIG.  43.    Surveyors'  cross. 


and  fix  D  on  line  A-B  by  intersection.  Then  by  similar 
triangles  C-D  =  D-E  and  the  angle  ADC  is  a  right  angle. 
By  sighting  from  E  to  C  the  line  connecting  C  and  D  may 
be  staked  out  on  both  sides  of  the  river. 

14.  To  erect  a  perpendicular  with  a  surveyors'  cross.  - 
The  surveyors'  cross  was  once  a  common  instrument, 
made  and  sold  by  all  instrument  manufacturers.  Several 
patterns  were  used.  Today  few  dealers  sell  this  instrument 
but  it  is  very  useful  in  chain  surveys 
when  no  compass  or  transit  is  avail- 
able.  In  Fig.  44  a  homemade  cross 
is  illustrated.  A  piece  of  wood  2  ins. 
thick  and  6  ins.  square  has  two  cuts  in 
it  normal  to  each  other  to  a  depth  of 
about  I  in.  It  is  fastened  to  a  Jacob 
staff  (a  pole  5  ft.  3  ins.  long,  about  \\ 
ins.  diameter,  pointed  and  shod  at  the 
lower  end  with  metal)  so  the  top  sur- 
face makes  a  right  angle  with  the  staff.  A  small  pocket 
level  is  used  with  the  cross,  generally  circular  in  form.  If 
a  circular  level  cannot  be  obtained  then  two  small  levels  are 
used,  one  parallel  with  each  cut. 

The  staff  is  set  in  the  ground  at  the  point  on  the  line  from 
which  the  perpendicular  is  to  be  set  off.  By  means  of  the 
level  the  staff  is  brought  to  a  truly  vertical  position  and  by 


I 


FIG. 


44.      Surveyors' 
cross. 


CHAIN   SURVEYING 


37 


sighting  forward  and  back  through  one  slit  this  slit  is  placed 
on  the  line.  Sighting  through  the  other  slit  stakes  may  be 
set  on  a  line  normal  to  that  on  which  the  cross  is  set.  It 
is  practically  impossible,  except  with  costly  tools,  to  make 
the  two  cuts  form  a  perfect  right  angle,  so  if  the  perpen- 
dicular line  is  to  be  set  off  with  more  than  ordinary  care 
the  principle  known  as  "double-centering"  must  be  used. 
After  sighting  through  the  slit  across  the  line  and  setting 
a  stake  turn  the  staff  180  deg.  and  get  the  first  slit  again  on 
line,  the  ends  being  reversed.  When  the  slit 
is  on  the  line  and  the  staff  vertical  sight  again 
at  the  stake  set  on  the  perpendicular  line. 
The  line  of  sight  will  fall  to  one  side  if  the 
two  slits  do  not  make  a  perfect  right  angle. 
The  correct  position  for  the  center  of  the 
stake  is  between  the  first  and  second  points 
fixed  by  the  two  sightings. 

With  such  a  primitive  instrument  a  difference  of  three 
or  four  inches  on  a  sight  of  more  than  100  ft.  will  not  be 
extraordinary.  By  "double  centering"  the  sights  and 
bisecting  the  points  the  line  will  be  exactly  located  if  proper 
care  is  used.  By  "double  centering"  the  error  is  doubled 
and  occurs  on  one  side  of  the  line  for  the  first  sight  and  on 
the  opposite  side  for  the  second  sight. 
The  reversal  of  observations  and 
doubling  of  errors,  to  obtain  a  mean 
(average)  is  the  underlying  principle 
'°'  of  all  adjustments  of  instruments. 
This  is  illustrated  in  Fig.  46. 

Let  A—B  be  a  line  with  which  line 
C-D  is  to  intersect  and  form  a  right 
angle.  The  cross  is  set  at  0  with 
a-b  on  the  line  A—B  and  sighting 

FiG.46.  Principleofdouble-    thr°USh   C^   the   P°intS    °'   a"d  ^' 


centering. 


are  set.  The  line  c— d  may  not  be 
truly  perpendicular  to  line  a-b  so  the 
cross  is  turned  180  deg.  and  by  sighting  through  c-d  the 
points  C"  and  D"  are  set.  No  further  explanation  is 
required  to  show  that  the  true  position  of  C  is  midway 
between  C'  and  C",  and  the  true  position  of  D  is  midway 
between  D'  and  D". 


38  PRACTICAL  SURVEYING 

Assuming  the  board  in  which  the  slits  are  cut  to  be  per- 
pendicular to  the  staff  the  board  will  be  level  when  the  staff 
is  truly  vertical.  If  the  board  is  not  level  all  angles  set  off 
by  using  it  will  be  too  small.  With  such  an  instrument 
(one  which  may  be  turned  180  deg.  only  by  turning  the 
staff)  no  correction  is  possible  for  errors  caused  by  lack  of 
horizontality  of  the  board.  With  transits  or  compasses, 
having  a  spindle  on  which  the  instrument  is  turned,  lack  of 
verticality  in  the  spindle  is  corrected  by  "  double  centering. " 

A  surveyors'  cross  made  by  a  scientific  instrument  maker 
sometimes  has  graduations  by  means  of  which  angles  of 
30  deg.,  45  deg.,  60  deg.  and  90  deg.  may  be  set  off.  A 
compass  or  transit,  however,  should  be  used  for  angles 
other  than  right  angles,  although  a  true  right  angle  is  the 
most  difficult  angle  to  turn. 

CHAIN  SURVEYS 

i .    To  survey  a  triangular  field.  —  On  the  left-hand  page 
of  the  field  book  set  down  the  lengths  of  the  lines  and  on 
the  right-hand  page  draw  a  diagram. 
This  is  shown  in  Fig.  47. 


Chains. 
A-B  =  20 

B-C  =  24 
C-A  =  18 


FIG.  47. 


2.  To  survey  a  field  with  any  number  of  sides.  —  Measure 
each  side  and  then  measure  tie  lines  to  divide  the  field  into 
triangles.  This  is  illustrated  in  Fig.  48. 

Sides.  Chains.      Tie  lines.         Chains. 

A-B 30.60  A-C 45.0 

B-C 20.40  A-D 35.0 

C-D 22 . 40  A-E 24. 20 

D-E 16.20 

E-F 13.50 

F-A 28 . 00  PIG.  48.   Tie  line  survey. 


CHAIN   SURVEYING 


39 


3.  To  survey  a  field  with  crooked  sides.  —  On  the  inside  of 
the  field  close  to  the  boundaries  lay  off  straight  lines  as 
long  as  possible.  From  these  lines  measure  perpendiculars 
to  intersect  the  boundaries  at  each  angle.  Run  tie  lines 
on  the  inside  to  divide  the  interior  survey  into  triangles. 
It  is  not  necessary  to  measure  the  boundaries,  the  tie  lines 
and  perpendiculars  sufficiently  fixing  the  positions  of  the 
corners.  This  is  illustrated  in  Fig.  49  and  the  following 
notes. 


Lengths. 

Offsets. 

Chains. 

Chains. 

A-B 

11.20 

I 

0.56 

at      5.40 

2 

....  1  .  4O 

8.26 

3 

0.36 

11.20 

4 

0.36 

B-C 

7.96 

i 

.  .  .  .O.2O 

at      2.36 

2 

0.36 

4.28 

3 

.  .  .  .0.96 

7.96 

4 

0.30 

C-D 

4.  68 

T"  V-"J 

0.30 

at      4.34 

D-E 

4.20 

at 

end  0.30 

E-F 

8.20 

i 

.  .  .  .0.40 

at      i  .  04 

2 

....0.86 

2.96 

3 

0.33 

5-88 

4 

.  .  .  .  i.oo 

8.20 

5 

.  .  .  .0.  12 

F-G 

7.96 

i 

1  .  20 

at     2.0 

2 

.  .  .  .0.24 

7.96 

3 

.  .  .  .0.  16 

G-A 

6  48 

\J  •    4J.W 

i 

0.80 

at      3.80 

6.48 

2 

.  .  .  .0.40 

Tie  lines. 

Chains. 

B-G 

71 

C-G 

•  / 

Q-I 

D-G 

•   7v5 

.  4-^ 

D-F 

8 

T"O 

.68 

FIG.  49.    Tie  line  and  offset 
survey. 


40 


PRACTICAL   SURVEYING 


FIG.  50.    Tie  line  and  offset 
survey. 


If  the  adjoining  owners  do  not  object,  time  and  labor 
may  sometimes  be  saved  by  running  lines  across  boundaries 
and  measuring  offsets  to  the  right  and  left.  Care  must  be 

taken  in  recording  the  notes. 
This  method  applied  to  the  last 
field  is  illustrated  in  Fig.  50. 

LOCATING   OBJECTS 

To  locate  an  object  on  line.  — 
The  line  is  A-B,  Fig.  51 ,  and  the 
object,    assumed    here   to   be   a 
house,    is   to   be  shown  on   the 
map. 

The  line  is  carried  past  the 
house  and  marks  placed  at  / 
and  g  where  it  meets  the  walls. 
From  some  point  a  on  line  meas- 
ure the  distances  a-c,  f-c  and  a-f.  On  the  other  side  of 
the  house  from  the  point  b  measure  b-d,  and  b-e.  When 
the  map  is  drawn  points  a  and  b,  f  and  g  are  marked  on  the 
line  A—B.  From  a  with  radius  a-c  describe  an  arc  and 
from  /  with  arc  f-c  describe  an  arc  intersecting  the  first  at  c. 
Join  c  to  /  and  prolong  the  line  through  e. 

From  b  with  radius  b— e  describe  an  arc  cutting  the  line 
c-f-e  at  e.  From  b  with  radius  b-d  describe  an  arc  through 
d.  From  e  draw  a  line  through  g  to  d.  Three  corners  cde 
of  the  building  are  now  located  and  the  remaining  corner 
is  at  the  intersection  of  lines  parallel  with  c-e  and  d-e, 
passing  through  c  and  d. 

(Note.  —  When  a  survey  is  made  for  the  purpose  of  draw- 
ing a  plat  or  map  and  objects  are  to  be  located  the  method 
of  route  surveys  should  be  followed.  The  first  station 
should  be  marked  o,  and  pegs  set  at  the  end  of  each  tape 
length.  If  the  surveyors'  unit  of  measure  is  used  the  sta- 
tions will  be  66  ft.  long  and  if  the  engineers'  unit  of  meas- 
ure is  used  the  stations  will  be  100  ft.  long.  Each  station 
contains  100  units  so  a  portion  will  be  designated  by  a 
decimal,  as  4.37  chains,  or  by  a  +  sign,  as  4  +  37.) 

When  the  line  is  marked  in  stations  all  objects  are 
located  by  reference  to  the  nearest  stations  or  plus  station. 


CHAIN  SURVEYING  41 

A  sketch  is  drawn  on  the  right-hand  page  of  the  book 
showing  the  shape  of  the  object  and  all  the  tie  lines.  Houses 
are  usually  rectangular,  which  simplifies  sketching. 

Fences  crossing  a  line.  —  Note  the  station  where  a  fence 
crosses,  and  from  points  on  line  25  ft.  each  side  of  the  fence 
measure  to  points  on  the  fence  25  ft.  from  the  line,  as  shown 


\5»gQ 


y 


FIG.  51.  FIG.  52. 

in  Fig.  52.  When  platting  this  sketch  A-B  is  the  line. 
From  Sta.  5  +  00  describe  an  arc  of  28  ft.  on  the  right  of 
the  line  and  from  Sta.  5  +  50  describe  an  arc  of  26  ft.  on 
the  left.  From  Sta.  5  +  26  describe  arcs  with  radius  of 
25  ft.  intersecting  these  two.  A  line  drawn  through  these 
two  points  of  intersection  and  through  Sta.  5  +  26  repre- 
sents the  fence  in  position  and  direction. 

Objects  not  on  line.  —  In  Fig.  53  a  house  is  shown  at  one 
side  of  the  line.  At  a  a  stake  is  set  by  intersection,  to  show 
where  the  line  of  the  side  of  the  house  produced  strikes  the 
survey  line  A—  B.  At  b  and  c  perpendiculars  are  erected 
and  the  distances  a-b,  b-d  and  c—e  measured.  The  points 
are  easily  platted,  following  the  instructions  previously  given. 


2    53      3 
FIG.  53. 

In  Fig.  54  a  more  simple  method  is  shown.  Let  a,  b,  c 
and  d  be  four  stakes  on  the  line  from  which  measurements 
are  made  to  e  and  /. 


42  PRACTICAL  SURVEYING 

By  following  the  methods  given,  and  such  variations  as 
will  present  themselves  with  experience,  a  complete  sur- 
vey of  any  farm  or  tract  of  land  may  be  executed  with  a 
satisfactory  degree  of  accuracy.  Maps  can  be  made  and 
areas  found  without  the  use  of  compass  or  transit. 

MAKING   MAPS    (OR  PLATS)    OF   CHAIN   SURVEYS 

To  plat  the  survey  of  the  triangular  field,  first  draw  to 
any  scale  the  line  A-B,  20  chains  long.  From  B  with  a 
radius  of  24  chains  describe  an  arc  passing  through  C.  From 
A  with  a  radius  of  18  chains  describe  an  arc  intersecting 
the  first  arc  at  C.  Draw  lines  connecting  the  first  arc  at  C. 
Draw  lines  connecting  A  and  C  and  B  and  C. 

To  plat  the  second  field  draw  the  line  A-B.  From  A  with 
radius  A-C  describe  an  arc  intercepting  an  arc  described 
from  B  with  radius  B-C.  From  A  with  radius  A-D  de- 
scribe an  arc  intercepting  an  arc  described  from  C  with 
radius  C-D.  From  A  with  radius  A-E  describe  an  arc 
intercepting  an  arc  described  from  D  with  radius  D-E. 
From  E  with  radius  E-F  and  from  A  with  radius  A-F 
describe  arcs  intercepting  at  F. 

To  plat  the  third  field  draw  the  line  A-B  and  fix  the 
points  C,  G,  D,  E  and  F  by  intersection.  Measure  on  each 
line  the  distances  marked  and  set  off  the  perpendicular 
offset  lines.  Connect  the  ends  of  the  lines.  At  the  corners 
the  outside  lines  must  be  produced  to  intersect. 

The  tie  lines  are  measured  to  divide  fields  into  triangles 
in  order  to  plat  them  as  described.  If  the  maps  are  drawn 
to  a  large  scale  they  may  be  divided  into  triangles  or  quad- 
rilaterals, using  fine-pointed  hard  pencils.  The  lengths  of 
all  the  lines  are  then  measured  with  a  scale  and  the  area  of 
each  interior  figure  found.  The  sum  of  the  areas  will  be 
the  area  of  the  whole  field. 

PLANE   MENSURATION 

A  rectangle  is  a  four-sided  figure  with  the  opposite  sides 
parallel  and  each  angle  a  right  angle,  Fig.  55. 

A  rectangle  with  all  sides  equal  is  called  a  square. 
Area  of  a  rectangle  =  length  X  breadth,  or  A  =  Ib. 
Area  of  a  right-angled  triangle. 


CHAIN   SURVEYING 


43 


In  Fig.  56  two  right-angled  triangles  equal  in  size  are 
placed  as  shown,  thus  forming  a  rectangle.  Since  the  area 
of  a  rectangle  =  Ib  then  the  area  of  a  right-angled  triangle  = 


FIG.  55- 

|  Ib,  or,  using  the  letters  commonly  denoting  the  base  and 
altitude  of  a  triangle,  A  =  J  ab  =  — 

Area  of  any  triangle,  Fig.  57. 

(a)  Area  ABC  =  AreaABD  +  AreaBDC 

AD  Xh  ,  DCXh 


2 

(AD 


DQh      ACXh 


O      D 

fa) 


FIG.  57. 

(b)  Area  ABD  =  Area  ABC  -  Area  BCD 
=  AC  X  h      CD  Xh 

2  2 

=  (AC  -  CD)h  =  ADXh 

2  2 

ri      f      ii             r        *  •       i      base  X  height    .     1-1, 
Therefore  the  area  of  any  triangle  =  —     -  ,  the  height 

being  measured  on  a  line  normal  to  the  base. 


44  PRACTICAL  SURVEYING 

When  the  perpendicular  height  cannot  be  readily  ob- 
tained the  area  may  be  found  by  means  of  the  expression 

A  =  Vs(S-a)  (S  -b)(S  -c), 
where  5  is  half  the  sum  of  the  three  sides  =  a   '  C . 

2 

Example.  —  In  a  triangular  field  the  three  sides  are  as 
follows : 

A-B  =  20  chains, 

B-C  =  24  chains, 

C-A  =  1 8  chains. 
Find  the  area 


24  +  20+18  _ 
o  — —  31. 


5  -  a  =  31  -  24  =    7. 

5  —  b  =  31  —  20  =  ii. 

S  -  c  =  31  -  18  =  13. 

S(S-a)(S-b)(S-c)  =31  X7X  11X13  =  31,031. 

A/3 1, 03 1  =  176.16  sq.  chains  =  17.616  acres. 

PROBLEMS 

Compute  the  area  of  the  field  in  Fig.  48.  Find  the  area 
of  each  triangle  into  which  the  field  is  divided  by  tie  lines, 
add  these  areas  and  the  sum  will  be  the  area  of  the  field. 

Area   of  a  parallelogram. — A   par- 

\~~i j — ~7    allelogram  is  a  four-sided  figure  with 

|  /  j  /     opposite  sides  parallel  but  none  of  the 

j£ V        angles  are  right  angles. 

FIG     g  Let  Aj  B,  C,  D  be  a  parallelogram 

and  from  B  erect  a  perpendicular  touch- 
ing C-D  at  E.  From  A  erect  a  perpendicular  meeting 
C-D  produced  to  F. 

Then  ABEF  form  a  rectangle  and  the  Area  =  AF  X  AB. 

Since  AD  and  BC  are  parallel  to  each  other  FD  =  EC. 
Since  AB  and  CD  are  parallel  to  each  other  AF  =  BE. 
The  AAFD  =  ABEC,  therefore  the  area  of  the  rectangle 
ABEF  =  area  of  the  parallelogram  A  BCD. 

The  area  of  a  parallelogram  =  base  X  perpendicular  height. 


CHAIN  SURVEYING 


45 


Area  of  a  trapezium. — A  trapezium  is  a  quadrilateral 
(four-sided  figure)  with  no  parallel  sides. 

Divide  the  trapezium  into  triangles  and  find  the  sum  of  the 
areas  of  the  triangles. 


FIG.  59. 


FIG.  60. 


Area  of  a  trapezoid.  —  A  trapezoid  is  a  quadrilateral  with 
two  sides  parallel.  It  may,  or  may  not,  contain  two  right 
angles'  at  the  base. 

IST  CASE.  —  No  included  right  angles.  Divide  the 
trapezium  ABCD  into  two  triangles  ABC  and  ADC,  then 

Area  ABCD  =  Area  (ABC)  +  Area  (ADC) 

_AD Xh  ,  BC X  h 

~~T  ~~T 

(AD  +  BC)  h 


2ND  CASE.  —  The  two  parallel  sides  are  perpendicular  to 
the  base.  D 

H 


Area  =  AB  X  AD  +  BC  =  AB  X  EF. 


£ 

FIG.  61. 


The  area  of  a  trapezoid  =  half  the  sum 
of  the  parallel  sides  X  the  perpendicular 
distance  between  them. 

The  area  of  an  irregular  field  bounded 
by   straight    lines   may    be   obtained    by 
reducing    the   platted    figure   to    a   single    equivalent    tri- 
angle. 

Let  the  field  be  represented  by  the  figure  ABCD,  Fig.  62. 

Connect  BD  with  a  light  pencil  line  (using  a  hard  pencil 
with  chisel-edge  point).  Draw  CF  parallel  to  BD,  produc- 
ing AD  to  F. 


PRACTICAL  SURVEYING 


Draw  the  line  BF.  ABDC  =  ABDF  for  they  have  a 
common  base  BD,  and  a  common  altitude,  between  the 
parallel  lines  BD  and  CF. 

ABGD  is  common  to  ABDC  and  ABDF  and  when  sub- 


FIG.  63. 

% 

tracted  leaves  ADFG  which  is  added  to  the  figure  to  replace 
ABCG  of  equal  area  which  is  subtracted  from  it. 

AABF  =  Trapezium  AB CD. 
Draw  the  perpendicular  BE. 

AFX  BE 
Area=  -  —  • 


,Fig.  63  shows  the  principle  extended  to  obtain  the  area 
of  a  five-sided  field. 

A            FGXCH 
Area=       — 

This  method  is  general  (applicable  to  fields  having  any 
number  of  sides). 

Problem.  —  Plat  the  field  shown 
in  Fig.  49.  Compute  the  areas 
of  the  triangles  formed  by  the 
tie  lines.  The  irregular  pieces 
around  the  edges  are  to  be  com- 
puted as  triangles  or  as  trape- 
zoids,  the  sum  of  the  areas  of 
all  the  small  divisions  being  the  area  of  the  field. 

Another  method  is  to  carefully  plat  the  survey  of  the 
field  and  draw  averaging  lines  through  the  boundaries,  thus 
reducing  the  field  to  a  trapezoidal  shape  (Fig.  64). 


FIG.  64. 


CHAIN  SURVEYING 


47 


The  small  pieces  on  either  side  of  the  averaging  lines  must 
balance  in  area. 

Area  of  a  field  bounded  by  a  curved  line. 

Trapezoidal  rule.  —  From  a  base  line  erect  perpendiculars 
dividing  the  field  into  an  even  number  of  strips,  the  inter- 
vals between  the  perpendiculars  (or-  ^ 
dinates)  being  equal.  The  accuracy 
of  the  computation  of  area  is  fixed  A[._|. 
by  the  number  of  ordinates.  The 
general  rule  is  to  have  them  so  close 
together  that  each  section  of  the 
intercepted  curve  may  be  considered 
to  be  practically  straight. 

The  sum  of  the  lengths  of  the  ordi- 
nates, divided  by  the  number  of  ordi- 
nates =  mean  ordinate. 

Area  =  mean  ordinate  X  length. 

The    operation    is    shortened    by 
adding  to  half  the  sum  of  the  lengths 
of  the  two  end  ordinates  the  sum  of  the  lengths  of  the  in- 
terior ordinates  and  multiplying  by  the  width  of  one  strip. 

This  rule  may  be  expressed  as  follows: 


FIG.  65. 


Let 


then 
A  = 


=  area, 

,5  =  width  of  strip, 
h  =  ordinate  (height), 

.  .  .  hn=  each  ordinate,  the  subscript  denoting  the 
particular  ordinate,  the  subscript  n 
indicating  the  last  ordinate, 

n  —  I  =  number  of  spaces  when  n  =  number  of 
ordinates, 


kn-l] 


(Note.  —  When  the  curve  touches  the  base  at  one  end  or 
at  both  ends  the  length  of  the  end  ordinate  =  o.) 


Assume  hi  =  o  and  hn  =  6  then 
Assume  h\  — 


hi  -f  hn      0  +  6 


=  3- 


,       o  -f-  o 
and  hn  =  o  then  -     -  =  o. 


48  PRACTICAL  SURVEYING 

To  pass  a  circle  through  three  points  not  lying  in  a  straight 
line. 

Let  A,  B,  C  be  the  points.     Join  A  to  B  and  B  to  C. 
Erect  a  perpendicular  from  the  middle  of  AB  and  also  from 
the  middle  of  BC,  the  perpendiculars  intersecting  at  some 
point  O.    From  O  as  a  center  with  a  radius 
=  OA  describe  a   circle  which   will   pass 
through  A ,  B  and  C. 

Thomas  Simpson,  Professor  of  Mathe- 
matics, Royal  Military  Academy,  Wool- 
wich, England;  born  Aug.  10,  1710;  died 
May  14,  1761;  was  the  author  of  a  rule 
FIG.  66.  for  finding  the  area  of  an  irregular  figure, 

used  in  ship-building  calculations.  The 
mathematical  proof  can  only  be  followed  by  students  who 
have  completed  a  college  course  in  mathematics,  but  it  is 
not  necessary  to  study  the  demonstration  in  order  to  use 
the  rule.  It  is  based  on  the  assumption  that  through  the 
ends  of  three  equidistant  ordinates  we  can  draw  arcs  of 
parabolas,  approximating  to  the  curve  between  these  or- 
dinates, and  a  series  of  arcs  can  be  thus  drawn  to  fit  the 
boundary  of  any  irregular  figure  with  a  curved  outline. 

Simpson1  s  Rule.  — The  figure  is  divided  by  an  odd  number 
of  perpendiculars  into  an  even  number  of  strips  of  equal  width . 
To  the  sum  of  the  lengths  of  the  first  and  last  ordinate 
add  twice  the  sum  of  the  lengths  of  the  remaining  odd  ordi- 
nates and  four  times  the  sum  of  the  even  ordinates.  Mul- 
tiply by  one-third  the  width  of  one  strip. 

Using  the  notation  given  for  the  trapezoidal  rule,   the 
area  by  Simpson's  rule  may  be  expressed  as  follows: 

A  =  - 


Examples.  —  Find  the  areas  of  the  following  irregular 
surfaces  by  the  trapezoidal  rule  and  by  Simpson's  rule. 
Simpson's  rule  is  the  more  accurate. 

(Note.  —  If  there  are  nine  ordinates,  the  first,  third, 
fifth,  seventh  and  ninth  ordinates  are  the  odd,  and  the 


CHAIN   SURVEYING  49 

second,   fourth,  sixth  and  eighth  are  the  even,  ordinates 
referred  to  in  the  rule.) 

1.  Length  of  base  20  feet.     Ordinates  =  10,  16,  14,  n, 
1 6  feet. 

2.  Length  of  base  325  links.     Ordinates  o,  25,  38,  51,  64, 
TO,  83,  96  links.     Here  we  have  a  curve  cutting  the  base  at 
one  end. 

3.  Length  of  base  252  links.     Ordinates  =  o,  24,  36,  42, 
54,  67,  76,  58,  49,  33,  19,  o.     Here  the  curve  cuts  the  base 
at  both  ends. 

4.  Length  of  base   1260  links.     Ordinates  =  364,   396, 
418,  453,  512,  554,  578  links. 

5.  Length  of  base  2364  links.     Ordinates  =  o,  335,  417 
432,  524.587,  642,  758  links. 

Area  by  weighing.  —  Plat  the  field  accurately  on  a  thick 
piece  of  paper  and  cut  it  out  carefully  with  a  sharp  knife. 
On  the  same  sheet  draw  a  square  or  rectangle  of  predeter- 
mined area  to  the  same  scale  and  cut  out.  The  more  nearly 
the  area  of  the  field  and  regular  figure 
agree  the  more  accurate  the  method.  a |  b 

On  a  light  rod  hang  the  cut-out  field  rf\  ~~~ 

(A)  at  one  end  and  the  regular  figure  ^^ 

(B)  at  the  other  end.     Suspend  the  rod  pIG  6? 
on  a  thin  wire  or  over  a  pin  and  move 

to  right  or  left  until  it  remains  in  a  horizontal  position, 
showing  that  A  balances  B. 

„,  .  A       A  rea  of  B  X  length  be 

The  area  of  A  =  -  -  • 

length  ac 

Area  by  planimeters. — A  planimeter  is  an  instrument 
used  for  obtaining  the  areas  of  irregular  figures.  There 
are  several  forms  and  full  instructions  for  use  are  given 
with  each  one  sold.  Pictures  with  descriptions  are  given 
in  the  catalogues  of  instrument  dealers. 

A  form  of  planimeter  that  the  student  can  make  is 
known  as  the  "hatchet"  planimeter,  because  one  end 
slightly  resembles  a  hatchet  blade.  Some  writers  call  it 
the  "jack-knife"  planimeter  because  it  resembles  a  knife 
with  a  projecting  blade  at  each  end.  For  demonstrating 
purposes  a  pocket  knife  is  a  fair  substitute  for  a  properly 
made  instrument  of  this  type. 


PRACTICAL  SURVEYING 


A  piece  of  wire  is  bent  at  A  and  B.  On  one  leg  is  fastened 
a  weight  F  and  the  lower  end  is  flattened  to  a  shape  which 
gives  the  instrument  the  name  of  "hatchet"  planimeter. 
At  E  on  the  hatchet  blade  end  make  a  mark.  The  other 


I 


FIG.  68.    Hatchet,  or  jack-knife,  planimeter. 

leg  has  a  sleeve  c,  in  which  the  wire  may  move  freely  but 
not  too  loosely.  The  lower  end  of  the  leg  AD  is  brought 
to  a  point. 

The  distance  between  D  and  E  may  be  any  length  but 
is  usually  some  even  number  of  inches,  preferably  a  decimal 
division  such  as  5,  10,  etc.  Some  planimeters  of  this  type 
have  an  adjustable  arm  AB  so  the  length  may  be  altered, 
enabling  the  instrument  to  be  used  for  comparatively  large, 
as  well  as  for  small,  drawings. 

Estimate  as  closely  as  possible  the  position  of  the  center 
of  the  area  of  the  drawing,  Fig.  69,  and  mark  it  0,  drawing 
through  it  the  lines  AB  and  CD  normal  to  each  other. 

The  accuracy  of  the  work  de- 
Of  pends  upon  the  accuracy  with 

which  the  point  o  fixes  the 
center  of  area  (center  of  grav- 
ity). Place  the  point  D  of 
the  planimeter  at  O  and  the 
mark  E  on  the  line  CD,  press- 
ing the  hatchet  edge  into  the 
paper  slightly  to  fix  the  posi- 
tion of  the  mark.  Holding 
the  sleeve  C  move  the  point 
D  along  the  line  AB  to  A. 
Then  go  to  the  right  carefully  tracing  the  perimeter  with 
the  point  until  A  is  reached,  and  the  point  again  follows 
the  line  AB  from  A  to  O.  The  hatchet  edge  will  be  close 
to  the  starting  point  on  the  line  CD  and  should  be  pressed 
into  the  paper  to  mark  the  new  position  of  the  point  E. 
Measure  the  distance  between  the  original  and  final  posi- 
tions of  E  normal  to  line  CD  and  call  this  a\. 


• — a 


CHAIN  SURVEYING  51 

2nd.  Place  D  at  O  and  E  on  the  line  CD  near  D,  which  is 
equivalent  to  changing  the  position  of  drawing  180  deg. 
Move  from  O  to  A  and  move  the  point  around  the  perim- 
eter to  the  left.  Call  the  distance  between  the  first  and 
final  positions  of  R  on  this  second  operation  0%. 

Area  =  length  of  DE  X  ai  +  a*. 


3rd.  Closer  results  may  be  obtained  by  next  placing 
the  planimeter  on  the  line  AB  with  E  near  A.  Move  D 
along  OD  towards  D  and  trace  the  perimeter  to  the  right. 
Call  the  distance  between  the  first  and  final  positions  of 
E  on  this  third  operation  a3. 

4th.  Place  D  at  O  with  E  on  the  line  AB  near  B.  Move 
towards  D  on  line  OD  to  the  perimeter  and  trace  the  perim- 
eter to  the  left.  Call  the  distance  between  the  first  and 
final  position  of  E,  on  this  fourth  operation  a*. 

Area  =  length  of  DE  X  ai  +  a*  +  a*  +  a*. 


To  become  expert  in  the  use  of  a  planimeter  the  be- 
ginner should  check  plats  of  regular  figures,  the  areas  of 
which  may  be  readily  found  by  common  methods. 

THE  USE  OF  SQUARED  PAPER 

Figures  may  be  platted  on  squared  paper  and  the  in- 
cluded squares  counted.  The  sum  multiplied  by  Jhe  area 
of  one  square  =  area  of  figure. 

CIRCLES 

The  perimeter  (boundary)  of  a  square  f 
with  side  d  =  4  d. 

The  perimeter  (circumference)  of  a  ! 
circle  with  diameter  d  =  *?-  d  =  3}  d  =  7 
3.1416^. 

The  area  of  a  square  with  side  d 


FlG- 


PRACTICAL  SURVEYING 


The  area  of  a  circle  with  diameter  d  =  —        -  =  0.7854  d2. 

4 

A  rea  of  a  ring. 
Let  D  —  outer  diameter, 
d   =  inner  diameter, 
A  =  0.7854  £>2- 0.7854^ 
=  0.7854  (D*-d2). 

An  excellent  text  on  mensuration  is 
"  Practical  Mathematics  "  by  Knott  & 
Mackay  ($2.00). 


3456     78 


STAKING   OUT  WORK 

To  stake  out  a  lot  for  grading.  —  Grade  stakes  on  lots  are 
generally  set  in  lO-ft.  or  25-1 1.  squares.  At  the  corner  of 
each  square  is  set  a  stake  about  one  foot  long  projecting 
4  ins.  or  6  ins.  above  the  surface,  and  on  this  stake  is  marked 
the  cut  or  fill. 

In  Fig.  72  assume  that  the  square  A  BCD  is  to  be  laid 
out  in  squares  or  rectangles.     From  A  in  any  convenient 
way  lay  off  the  line  AD,  perpendicular 
to  AB,  setting  the  stakes  and  marking 
them  Oa,  Ob,  Oc,  etc.      From  B  set  off 
the  line  B  C,  perpendicular  to  BA,  set- 
ting .the  stakes  and  marking  them  8  a, 
8  b,  8  c,  etc. 

Measure  from  A  toB,  setting  the  stakes 
and  marking  them  I  a,  2  a,  3  a,  etc.  „..„.._,„ 

Measure   from   D    to    C,   setting  the 
stakes  and  marking  them  o  i,  I  i,  2  i,  etc.  IG'  72> 

A  4-ft.  lath,  with  a  cloth  tied  near  the  top,  is  set  on  the 
line  BC  at  each  stake  and  also  on  the  line  DC  at  each 
stake.  An  observer  stands  at  I  a  and  sights  towards  I  d. 
Another  observer  stands  at  o  b  and  sights  towards  8  b. 

A  helper  with  a  bag  of  stakes  and  a  line  pole  goes  along 
the  line  b  and  is  lined  in  by  the  two  observers  by  the  in- 
tersection process,  the  observer  on  the  b  line  remaining  in 
place,  the  observer  on  the  a  line  moving  towards  B  after 
each  stake  is  driven. 


CHAIN   SURVEYING 


53 


When  stake  7  a  is  set,  the  observer  at  o  b  moves  to  o  c 
and  the  helper  goes  to  the  c  line  and  sets  7  c.     He  and  the 
observer  on  the  a  line  work  back  to  the  I  line.     The  other 
observer  then  drops  to  the  d  line,   and  the 
work  proceeds  thus  until  all  the  stakes  are 
set.     In  Fig.  73  a  stake  is  shown,  every  stake 
being  marked  to  indicate  the  intersection  of 
the  two  lines  on  which  it  stands.     The  lower 
figures  +7.3  are  put  on  after  the  elevations 
are  taken.  FIG.  73. 

The  -f-  sign  indicates  a  cut,  the  surface  of 
the  ground  at  this  stake  being  7.3  feet  above  grade.     A 
—  sign  indicates  a  fill. 

To  stake  out  a  building.  —  The  3,  4,  5  method  for  erect- 
ing perpendiculars  is  generally  used  by  contractors  when 


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FIG.  74.     Staking  out  a  building. 

staking  out  buildings  without  an  instrument  for  turning 
angles. 

In  Fig.  74  the  property  line  AB  is  assumed  to  be  staked 


54 


PRACTICAL  SURVEYING 


and  the  location  of  the  building  settled.     The  heavy  lines 
show  the  outline  of  the  building. 

At  I  and  n  stakes  are  set,  the  distance  between  being 
10  ft.  more  than  the  width  of  the  building,  each  stake 
being  5  ft.  from  the  building  line.  A  chalk  line  is  stretched 
tightly  from  nails  driven  to  mark  the  exact  points  in  the 
tops  of  the  stakes  and  the  points  2  to  10  inclusive  set  and 
nails  are  driven  to  mark  them. 

At  I  and  n  perpendicular  lines  (i  ...  21  and  n  to 
21 0  are  set  out,  the  stakes  numbered  21  being  5  ft.  from 

the  building  line.  At  2 1  check 
measurements  are  made  to 
make  it  certain  that  perfect 
right  angles  have  been  turned. 
Lines  are  stretched  from  I 
to  21  and  from  n  to  21',  the 
intermediate  points  12  to  20 
inclusive  measured  off  and 
set  with  nails  marking  the 
exact  locations. 
When  a  stake  is  used  the  top  is  usually  I  or  2  ft.  above 
the  ground  and  the  nail  projects  an  inch  or  so  above  the 
top  so  a  cord  may  be  tied  to  it.  If  a  number  of  points  fall 
within  a  space  of  10  or  12  ft.  a  couple  of  posts  are  driven 
to  which  a  board  is  spiked  on  the  line  and  nails  are  driven 
in  the  top  edge  of  the  board. 


FIG.  75.    Batter  boards. 


Batter  boards  at  corner. 


The  dotted  lines  represent  cords  stretched  between  nails 
and  the  points  of  intersection  fix  the  building  lines.  The 
stakes  being  at  least  5  ft.  away  from  the  building  are 
safe  from  disturbance  during  the  excavating  period.  It  is 


CHAIN  SURVEYING  55 

never  necessary  to  have  all  the  lines  in  place  at  one  time. 
Two  men  can  stretch  line  2  .  .  .  2'  and  then  hold  a  line 
on  1 8  .  .  .  1 8'  while  a  third  man  drives  a  stake  at  F. 
Moving  to  17  ...  17'  a  stake  is  driven  at  G.  Line 
2  ...  2'  is  moved  to  3  ...  3',  the  cross  line  held  on 
16  .  .  .  16'  while  //  is  driven  and  on  19  ...  19'  while  E 
is  driven. 

The  corner  line  guides  are  often  arranged  as  shown  in 
Fig.  76,  and  the  cords  are  tied  around  the  boards,  as  the 
pull  on  a  long  line  will  bend  a  small  nail,  for  all  lines  must 
be  taut. 

THE  USE  OF  A  TABLE  OF  SQUARES 

Surveyors  should  become  accustomed  to  the  use  of 
labor-saving  tables  and  diagrams.  The  table  here  given  is 
very  old  and  of  great  value. 

Find  the  square  of  138. 

The  square  of  138  =  138  X  138  =  I382.  Looking  in 
the  column  headed  No.  find  138.  In  the  column  headed 
Square  find  19,044  opposite  138. 

Then  I382  =  19,044. 

Find  the  cube  of  138. 

The  cube  of  138  =  138  X  138  X  138  =  I383.  Looking 
in  the  column  headed  No.  find  138.  In  the  column  headed 
Cube  find  2,628,072  opposite  138.  Then  I383  =  2,628,072. 

Proceeding  in  a  similar  manner  the  square  root  of 
138  =  V^S  =  11.747344  and  the  cube  root  of  138  = 
^138  =  5.167649. 

The  square  root  of  any  number  is  one  of  two  equal  factors 
which  multiplied  together  will  produce  that  number. 
Thus  5  is  the  square  root  of  25,  for  25  =  5  X  5- 

The  cube  root  of  any  number  is  one  of  three  equal  factors 
which  multiplied  together  produce  that  number.  Thus  5  is 
the  cube  root  of  125,  for  125  =  5  X  5  X  5. 

The  fourth  root  of  any  number  is  one  of  four  equal 
factors  which  multiplied  together  produce  that  number. 
Thus  5  is  the  fourth  root  of  625,  for  625  =  5X5X5X5. 

The  sixth  root  of  any  number  is  one  of  six  equal  factors 
which  multiplied  together  produce  that  number.  Thus  5 
is  thesixth  root  of  1,953, 125,  for  1,953,125  =  5  X5  X5  X5  X 
5X5- 


56  PRACTICAL  SURVEYING 

Let       a  =  any  number,  then 

a2  =  a  X  a, 

and          a*  =  aXaXaXa. 
.'.  a4  =  a2  X  a2  =  a2+2. 

a3  =  aXaXa, 

and          a*  =  aXaXaXaXaXa. 
.'.  a6  =  a3  X  a3  =  a3+3. 

The  exponent  =  the  sum  of  the  times  the  number  is 
used.  It  follows  that: 

a5  =  a  X  a  X  a  X  a  X  a, 

=  a2  X  a3  =  a2+3, 
and          a7  =  a2  X  a2  X  as  =  a4  X  a3  =  a4+3. 

A  knowledge  of  the  law  of  exponents  permits  the  use  of 
the  table  to  find  the  fourth  and  higher  powers. 

Thus  to  find  the  fourth  power  of  any  number,  first  find 
the  square.  Using  the  square  as  a  number  find  its  square. 
Find  the  fourth  power  of  30. 

302  =  900, 
9002  =  810,000  =  304  =  302  X  3°2  =  3<>  X  30  X  30  X  30- 

The  sixth  power  is  found  by  using  the  cube.  Find  the 
sixth  power  of  7. 

7*  =343, 

3433  =  40,353,607  =  76  =  73X73  =  7X7X7X7X7X7. 

To  find  the  fifth  power  of  a  number  find  the  square  and 
the  cube  of  the  number  and  multiply. 
Find  the  fifth  power  of  2. 

22  =  2  X  2  =  4, 

23  =  2X2X2  =  8, 

25  =  22X23  =  4X8  =  32   =2X2X2X2X2. 

To  find  the  seventh  power  of  a  number  multiply  the 
third  power  by  the  fourth  power. 
Find  the  seventh  power  of  2. 

23  =  2X2X2  =  8, 

24  =  2  X  2  X  2  X  2  =  16, 

27  =  23X24  =  8Xi6  =  128  =  2X2X2X2X2X2X2. 


CHAIN   SURVEYING 


57 


In  Fig.  77  Aa  =  —  and  ABCD  form  a  square. 


Aa'  = 


a'e  =  ed  =  Ad.     The  square  ABCD  has  an  area  four  times 
that  of  the  square  Aa'ed,  having  sides  one-half  the  length 


FIG.  78. 

of  the  larger  square.  Similarly  in  Fig.  78  the  square  ABCD 
has  an  area  nine  times  as  large  as  the  area  of  the  square 
Aabc,  having  sides  one-third  the  length  of  the  larger  square. 

The  table  contains  no  numbers  larger  than  1000,  there- 
fore the  principle  just  illustrated  must  be  used  when  the 
square  of  a  large  number  is  wanted. 

Rule.  —  Divide  by  a  number  giving  a  quotient  contain- 
ing less  than  four  figures.  Multiply  the  square  of  the 
quotient  by  the  square  of  the  divisor. 

Example.  —  Find  the  square  of  1220. 


1220 
10 


122. 


I222  =   14,884. 
I2202  =   I222  X  I02  =   14,884  X  100  =   1,488,400. 

Find  904. 

902  =  8100. 
810. 


8100 


10 

8io2  =  656,100. 
904  =  8io2  X  io2  =  656,100  X  100  =  65,610,000. 

Find  954. 

952  =  9025. 
9025 

V  =  902-5. 


58  PRACTICAL  SURVEYING 


25  250 

36.I2  =  1303-21. 
2502  =  62,500. 
954  =  36.I2  X  2502  =  1303.21  X  62,500  =  81,450,625. 

In  the  last  example  the  method  of  handling  squares  of 
decimal  numbers  is  plainly  illustrated. 


The  square  of  361  =  130,321. 
The  square  of  36.1  =  1303.21. 
The  square  of  3.61  =  13.0321. 


To  extract  the  square  root  of  any  number.  —  The  square 
root  of  each  number  less  than  1000  is  found  in  the  column 
headed  Square  Root.  If  the  number  is  greater  than  1000 
find  it  in  the  column  headed  Square  and  the  square  root 
will  be  found  in  the  column  headed  No. 

To  extract  the  cube  root  of  any  number.  —  The  cube  root 
of  each  number  less  than  1000  is  found  in  the  column 
headed  Cube  Root.  If  the  number  is  greater  than  1000 
find  it  in  the  column  headed  Cube  and  the  cube  root  will 
be  found  in  the  column  headed  No. 

When  the  cube  is  wanted  of  a  number  larger  than  1000 
it  will  be  best  to  use  logarithms. 

Barlow's  Table  Book  contains  squares,  square  roots, 
cubes  and  cube  roots  for  all  numbers  from  I  to  10,000,  and 
is  sold  by  all  dealers  in  scientific  books.  No  books  are  in 
general  circulation  containing  squares  of  higher  numbers 
because  of  the  small  demand,  although  tables  of  squares 
of  all  numbers  up  to  100,000  were  once  printed. 

8    a  =  altitude, 
b  =  base, 
c  =  hypothenuse, 

C  —   v  CL      i     0  *         * 


c    b  =  V(c  +  a)  X  (c  -  a)  =  V  c2  -  a2, 
FIG.  79.  a  =  V(c  +  b)  X  (c  -  b)  =  Vc2  -  b2. 

The  above  formulas  should  be  memorized  for  they  are 
very  useful  to  surveyors. 


CHAIN  SURVEYING  59 

The  formulas  for  rinding  the  base  and  altitude  of  a  right- 
angled  triangle  illustrate  the  algebraic  theorem 

The  product  of  the  sum  and  difference  of  two  quantities  = 
the  difference  of  their  squares. 

Examples.  —  Use  the  table  of  squares. 

(i)        « -> 

6  =  4.     Find  c. 

a2  =  32=    9 

1 6  . 

&2  =  42  =  ~         c=V25  =  5 

In  the  column  headed  Squares  find  25,  and  in  column 
headed  No.  find  5. 

Another  method  is  to  find  25  in  the  column  headed  No. 
and  in  the  column  headed  Square  Root  find  5. 

(2)  c  =  35. 

b  =  28. 
c2  =  352  =  1225 


441 

When  either  number  contains  more  than  three  figures 
both  numbers  must  be  divided  by  a  number  that  will  re- 
duce them  to  less  than  four  figure  values. 

Algebraically  and  geometrically  we  can  prove 
The  value  of  a  ratio  is  not  altered  when  both  terms  are 
multiplied  or  divided  by  the  same  quantity. 

3-2    y    1  "2    y    C  1    y    *7 

_6^6_6^o_6^i     e£C 


4      4X3      4X5      4X7' 

Using  the  new  values  proceed  as  before.  When  the 
square  root  is  found  multiply  by  the  number  used  as  a 
divisor.  The  result  will  be  the  same  as  though  the  original 
values  had  been  used. 

We  can  prove  by  geometry 

Triangles  which  have  two  angles  equal  each  to  each  have 
their  sides  proportional. 


60  PRACTICAL    SURVEYING 

Example. 

a  =  117.28  ft. 

b  =  92.20  ft.     Find  c. 

Divide  by  a  common  divisor. 

4|ii7.28         92.20 

4129.32         23.04  Divisor  =  4  X  4  =  16 

7-33  576  + 

7-332  =  537289 

5-762  =  33.1776 

86.9065 

From  table  86.8624  =  9.32 

0.0441          1 6 

5592 

932 
149.12 
add  0.05 

149.17  =  c. 

Squaring  the  lengths  of  the  two  sides  and  extracting 
the  square  root  of  the  sum  by  arithmetic  the  length  of 
c  =  149.182.  The  difference  is  closer  than  ordinary  work 
in  the  field. 

No  rational  reason  can  be  given  for  adding  the  remainder 
0.0442  (0.05)  but  we  know  the  root  lies  between  9.32  and 
9.33,  and  experience  has  shown  that  when  the  second 
significant  figure  in  the  remainder  is  increased  by  I  and 
the  remainder  then  added,  the  final  error  is  very  small. 
With  a  table  of  squares  of  numbers  up  to  10,000  more 
exact  results  can  be  obtained. 

SLIDE  RULE 

The  slide  rule  has  become  an  indispensable  tool  for 
engineers,  and  should  be  used  by  every  surveyor.  The 
principle  of  the  slide  rule  is  simple  although  some  very 
complicated  forms  are  manufactured.  Full  instructions 
for  use  accompany  each  instrument.  Purchase  only  from 


CHAIN  SURVEYING  6l 

a  firm  of  high  reputation  and  test  the  graduations  care- 
fully to  see  that  the  A  scale  coincides  with  the  B  scale,  and 
the  C  scale  with  the  D  scale. 

The  best  form  for  the  use  of  a  surveyor  has  the  ordinary 
Mannheim  graduations  with  a  reciprocal  scale  so  three 
numbers  can  be  multiplied  at  one  setting.  For  office  use 
a  i6-in.  slide  rule  is  best.  For  ordinary  use  either  an 
8-in.  or  lo-in.  rule  will  be  found  satisfactory.  The  man  who 
is  a  slave  to  the  slide  rule  carries  one  5  ins.  long  in  his 
pocket. 

MULTIPLYING  TABLES 

Crelle's  Multiplying  and  Dividing  Tables  ($5.00)  should 
be  in  the  office  of  every  man  who  has  to  do  much  figuring. 
On  a  large  survey  the  time  saved  will  pay  for  the  book, 
and  mistakes  can  occur  only  through  grave  carelessness. 


62  PRACTICAL  SURVEYING 

SQUARES,  CUBES,  SQUARE  ROOTS  AND  CUBE  ROOTS 


Nos. 

Squares. 

Cubes. 

Square 
root. 

Cube 
root. 

Nos. 

Squares. 

Cubes. 

Square 
root. 

Cube 
root. 

i 

t 

i 

I.  OOO 

.000 

51 

26  oi 

132  651 

7.141 

3.708 

2 

4 

8 

I.4I4 

.260 

52 

2704 

140  608 

7.  211 

3-733 

3 

27 

1-732 

•  442 

53 

2809 

148  877 

7.280 

3-756 

4 

16 

64 

2.  OOO 

•  587 

54 

29  16 

157  464 

7.349 

3.78o 

5 

25 

125 

2.236 

.710 

55 

3025 

166  375 

7.416 

3-803 

6 

36 

216 

2.449 

.817 

56 

3136 

175616 

7.483 

3-826 

7 
8 

S 

343 

512 

2.646 
2.828 

.913 

2.000 

57 
58 

3249 
3364 

185  193 
195  112 

7.550 
7.616 

3.849 
3.871 

9 

81 

729 

3.000 

2.080 

59 

348i 

205  379 

7.681 

3-893 

10 

I  00 

I  000 

3.162 

2.154 

60 

3600 

216  ooo 

7.746 

3.915 

ii 

I  21 

I  331 

3.317 

2.224 

61 

3721 

226981 

7.810 

3-937 

12 

144 

I  728 

3-464 

2.289 

62 

3844 

238  328 

7.874 

3-958 

13 

169 

2  197 

3-606 

2-351 

63 

3969 

250  047 

7.937 

3-979 

14 

196 

2744 

3-742 

2.410 

64 

4096 

262  144 

8.000 

4.000 

15 

225 

3375 

3.873 

2.466 

65 

4225 

274  625 

8.062 

4.021 

16 

256 

4096 

4.000 

2.520 

66 

4356 

287496 

8.124 

4.04T 

17 

289 

4913 

4-123 

2.571 

67 

4489 

300763 

8.185 

4.062 

18 

324 

5832 

4-243 

2.621 

68 

4624 

314  432 

8.246 

4.082 

19 

36i 

6859 

4-359 

2.668 

69 

476i 

328509 

8.307 

4.102 

20 

400 

8  ooo 

4.472 

2.714 

70 

4900 

343000 

8.367 

4.121 

21 

441 

9  261 

4.583 

2.759 

71 

5041 

357  9" 

8.426 

4.141 

22 

484 

10648 

4.690 

2.802 

72 

5184 

373  248 

8.485 

4.160 

23 

529 

12  167 

4-796 

2.844 

73 

5329 

389  017 

8.544 

4-179 

24 

576 

13824 

4.899 

2.885 

74 

5476 

405  224 

8.602 

4.198 

25 

625 

15625 

5.000 

2.924 

75 

5625 

421  875 

8.660 

4.217 

26 

676 

17576 

5-099 

2.963 

76 

5776 

438  976 

8.718 

4-236 

27 

729 

19683 

5.196 

3.000 

77 

5929 

456533 

8.775 

4-254 

28 

784 

21952 

5.292 

3-037 

78 

6084 

474  552 

8.832 

4-273 

29 

841 

24389 

5-385 

3.072 

79 

62  41 

493  039 

8.888 

4.291 

30 

9  oo 

27  ooo 

5-477 

3-107 

80 

6400 

512000 

8-944 

4.309 

31 

961 

29791 

5-568 

3.I4I 

81 

6561 

531  441 

9.000 

4-327 

32 

1024 

32768 

5.657 

3-175 

82 

6724 

551  368 

9-055 

4-345 

33 

1089 

35  937 

5-745 

3.208 

83 

6889 

571  787 

9.110 

4.362 

34 

ii  56 

39304 

5.831 

3.240 

84 

7056 

592  704 

9-165 

4.38o 

35 

1225 

42875 

5.916 

3-271 

85 

7225 

614  125 

9.220 

4-397 

36 

1296 

46656 

6.000 

3-302 

86 

7396 

636  056 

9-274 

4.414 

37 
38 

1369 

1444 

50653 
54872 

6.083 
6.164 

3-332 
3-362 

87 
88 

7569 
7744 

658  503 

63i  472 

9-327 
9-38i 

4-431 
4-448 

39 

1521 

59319 

6.245 

3-391 

89 

7921 

704  969 

9-434 

4.465 

40 

1600 

64  ooo 

6.325 

3.420 

90 

81  oo 

729  ooo 

9-487 

4.481 

41 

1681 

68  921 

6.403 

3.448 

91 

8281 

753  571 

9-539 

4.498 

42 
43 

1764 
1849 

74088 
79507 

6.481 
6.557 

3.476 
3.503 

92 
93 

8464 
8649 

778  688 
804  357 

9-592 
9.6l»4 

4.514 
4-531 

44 

1936 

85184 

6.633 

3-530 

94 

8836 

830  584 

9.695 

4-547 

45 

20  25 

91  125 

6.708 

3-557 

95 

9025 

857  375 

9-747 

4.563 

46 

21  16 

97  336 

6.782 

3.583 

96 

92  16 

884  736 

9-798 

4-579 

47 

22  09 

103  823 

6.856 

3.609 

97 

9409 

912  673 

9.849 

4-595 

48 

2304 

no  592 

6.928 

3.634 

98 

9604 

941  192 

9.900 

4.610 

49 

24  oi 

117  649 

7.000 

3.659 

99 

98  oi 

970  299 

995Q 

4.626 

50 

25  oo 

125  ooo 

7.071 

3-634 

100 

I  00  OO 

I    OOO  000 

10.000 

4-642 

CHAIN  SURVEYING  63 

SQUARES,  CUBES,  SQUARE  ROOTS  AND  CUBE  ROOTS     (Continued) 


Nos. 

Squares. 

Cubes. 

Square 
root. 

Cube 
root. 

Nos. 

Squares. 

Cubes. 

Square 
root. 

Cube 
root. 

101 

I  O2  OI 

i  030  301 

10.0499 

4-6570 

151 

2  28  01 

3  442  95i 

12.2882 

5.3251 

IO2 

10404 

i  061  208 

10.0995 

4-6723 

152 

23104 

3  5H  808 

12.3288 

5-3368 

103 

i  06  09 

i  092  727 

10.1489 

4.6875 

153 

23409 

3  58l  577 

12.3693 

5.3485 

104 

I  08  16 

i  124  864 

10.1980 

4.7027 

154 

23716 

3  652  264 

12.4097 

5.3601 

105 

I  10  25 

I  157  625 

10.2470 

4-7177 

155 

24025 

3  723  875 

12.4499 

5-3717 

106 

I  1236 

i  191  016 

10.2956 

4.7326 

156 

24336 

3  796  416 

12.4900 

5-3832 

107 

I  14  49 

i  225  043 

10.3441 

4-7475 

157 

24649 

3  869  893 

12.5300:  5-3947 

108 

i  16  64 

i  259  712 

10.3923 

4.7622 

158 

24964 

3  944  312 

12.5698    5.4061 

109 

I  18  81 

i  295  029 

10.4403 

4.7769 

159 

252  81 

4  019  679 

12.6095 

5.4175 

no 

I  21  OO 

i  331  ooo 

10.4881 

4-7914 

160 

25600 

4  096000 

12.6491 

5.4288 

III 

12321 

I  367  631 

10.5357 

4-8059 

161 

2  59  21 

4  173  281 

12.6886 

5-4401 

112 

I  25  44 

I  404  928 

10.5830 

4-8203 

162 

262  44 

4  251  528 

12.7279    5-4514 

H3 

i  27  69 

I  442  897 

10.6301 

4.8346 

163 

2  65  69 

4  330  747 

12.7671    5.4626 

114 

I  29  96 

I  481  544 

10.6771 

4.8488 

164 

26896 

4  410  944 

12.8062    5-4737 

US 

13225 

I  520  875 

10.7238 

4.8629 

165 

27225 

4  492  125 

12.8452 

5.4848 

116 

I  3456 

I  560  896 

10.7703 

4.8770 

166 

27556 

4  574  296 

12.8841 

5-4959 

117 

13689 

I  601  613 

10.8167 

4.8910 

167 

27889 

4  657  463 

12.9228 

5-5069 

118 

i  39  24 

I  643  032 

10.8628 

4.9049 

168 

2  82  24 

4  741  632 

12.9615 

5.5178 

119 

i  41  61 

i  685  159 

10.9087 

4.9187 

169 

2  85  61 

4  826  809 

13.0000 

5.5288 

120 

14400 

i  728  ooo 

10.9545 

4.9324 

170 

28900 

4  913  ooo 

13-0384 

5-5397 

121 

I  46  41 

i  771  561 

II.OOOO 

4.9461 

171 

29241 

5  ooo  211 

13-0767 

5.5505 

122 

14884 

i  815  848 

11.0454 

4-9597 

172 

29584 

5088448 

13-1149 

5.5613 

123 

151  29 

i  860  867 

11.0905 

4-9732 

173 

29929 

5  177  717 

13-1529 

5.5721 

124 

i  53  76 

i  906624 

II  -1355 

4.9866 

174 

30276 

5  268  024 

13-1909 

5.5828 

125 

15625 

i  953  125 

11.1803 

5.0000 

175 

30625 

5  359  375 

13.2288 

5-5934 

126 

15876 

2  000376 

11.2250 

5-0133 

176 

30976 

5  451  776 

13-2665 

5.6041 

127 

i  61  29 

2  048  383 

11.2694 

5-0265 

177 

31329 

5  545  233 

13-3041 

5.6147 

128 

16384 

2  097  152 

11-3137 

5-0397 

178 

3  16  84 

5  639  752 

13.3417 

5.6252 

129 

i  66  41 

2  146  689 

11-3578 

5-0528 

179 

32041 

5735339 

13-3791 

5.6357 

130 

i  6900 

2  197000 

11.4018 

5-0658 

180 

32400 

5  832  ooo 

13.4164 

5.6462 

131 

i  71  60 

2  248  091 

n.4455 

5.0788 

181 

3  27  61 

5  929  741 

I3.4536 

5-6567 

132 

i  7424 

2  299  968 

11.4891 

5.0916 

182 

33124 

6  028  568 

13.4907 

5-6671 

133 

2  352  637 

H.5326 

5-1045 

183 

33489 

6  128  487 

13.5277 

5.6774 

134 

i  79  56 

2  406  1O4 

11-5758 

5-II72 

184 

33856 

6  229  504 

13-5647 

5.6877 

135 

i  82  25 

2  460  375 

11.6190 

S.I299 

185 

34225 

6  331  625 

13-6015 

5.6980 

136 

18496 

2  515  456 

11.6619 

5.1426 

186 

34596 

6  434  856 

13-6382 

5.7083 

137 

18769 

2  571  353 

11.7047 

5.I55I 

187 

34969 

6  539  203 

13-6748 

5.7185 

138 

i  9044 

2  628  O72 

11-7473 

5-1676 

188 

35344 

6  644  672 

I3.7H3 

5-7287 

139 

I932I 

2  685  6l9 

11.7898 

S-lSoi 

189 

35721 

6  751  269 

13-7477 

5-7388 

140 

i  96  oo 

2  744000 

11.8322 

5.1925 

190 

3  61  oo 

6859  ooo 

13.7840 

5.7489 

141 

19881 

2  803  221 

II-8743 

5-2048 

191 

36481 

6  967  871 

13-8203 

5.7590 

142 

2  OI  64 

2  863  288 

11.9164 

5.2171 

192 

36864 

7  077  888 

13-8564 

5.7690 

143 

20449 

2  924  207 

H.9583 

5.2293 

193 

37249 

7  189  057 

13.8924 

5.7790 

144 
145 

2  0736 
2  1025 

2  985  984 

3  048  625 

I2.OOOO 
I2.04I6 

5.2415 
5  2536 

194 
195 

37636 
38025 

7  301  384 
7  4M  875 

13.9284 
13.9642 

5.7890 
5.7989 

146 

2  13  16 

3  112  136 

12.0830 

5.2656 

196 

38416 

7  529  536 

14.0000 

5.8088 

147 

2  1609 

3  i?6  523 

12.1244 

5.2776 

197 

38809 

7  645  373 

14.0357 

5.8186 

148 

2  1904 

3  241  792 

12.1655 

5-2896 

198 

39204 

7  762  392 

14.0712 

5.8285 

149 

2  22  01 

3  307  949 

12.2066 

5-3015 

199 

396oi 

7  880  599 

14.1067 

5.8383 

150 

2  25  00 

3375000 

12.2474 

5.3133 

200 

4  oo  oo 

8  ooo  ooo 

14.1421 

5-8480 

£4  PRACTICAL  SURVEYING 

SQUARES,  CUBES,  SQUARE  ROOTS  AND  CUBE  ROOTS     (Continued) 


Nos. 

Squares. 

Cubes. 

Square 
root. 

Cube 
root. 

Nos. 

Squares. 

Cubes. 

Square 
root. 

Cube 

root. 

201 

40401 

8  120  601 

14.1774 

5.8578 

251 

6  30  01 

15  813  251 

15  8430 

6.3080 

2O2 

40804 

8  242  408 

14.2127 

5-8675 

252 

6  .35  04 

16  003  008 

15.8745    6.3164 

203 

41209 

8  365  427 

14.2478 

5.8771 

253 

64009 

16  194  277 

15.9060!  6.3247 

204 

4  16  16 

8  489  664 

14.2829 

5.8868 

254 

6  45  16 

16  387  064 

15-9374    6.3330 

205 

42025 

8  615  125 

14.3178 

5-8964 

255 

65025 

16  581  375 

15.9687    6.3413 

206 

42436 

8  741  816 

14.3527 

5.9059 

256 

65536 

16  777  216 

i6.oooo|  6.3496 

207 

42849 

8  869  743 

14-3875 

5-9155 

257 

6  6049 

16  974  593 

16.0312    6.3579 

208 

43264 

8  998  912 

14.4222 

5.9250 

258 

66564 

17  173  512 

16.06241  6.3661 

209 

4368i 

9  129  329 

14.4568 

5-9345 

259 

6  7081 

17  373  979 

16.0935 

6.3743 

210 

4  41  oo 

9  261  ooo 

14.4914 

5-9439 

260 

6  76  oo 

17  576  ooo 

16.1245 

6.3825 

211 

4  45  21 

9  393  93i 

14.5258 

5-9533 

261 

68121 

17  779  58i 

16.1555 

6.3907 

212 

44944 

9  528  128 

14-5602 

5.9627 

262 

686  44 

17  984  728 

16.1864 

9.3988 

213 

45369 

9  663  597 

14-5945 

5-9721 

263 

691  69 

18  191  447 

16.2173 

6.4070 

214 

45796 

9800344 

14.6287 

5.9814 

264 

69696 

18  399  744 

16.2481 

6.4I5I 

215 

4  6225 

9  938  375 

14.6629 

5.9907 

265 

70225 

18  609  625 

16.2788 

6.4232 

216 

46656 

10  077  696 

14-6969 

6.0000 

266 

70756 

18  821  096 

16.3095 

6.4312 

21? 

47089 

10  218  313 

14.7309 

6.0092 

267 

7  1289 

19  034  163 

16.3401 

6.4393 

218 

47524 

10  360  232 

14.7648 

6.0185 

268 

71824 

19  248  832 

i  6  3707 

6.4473 

219 

47961 

10  503  459 

14.7986 

6.0277 

269 

72361 

19  465  109 

16.4012 

6.4553 

220 

48400 

10  648  ooo 

14.8324 

6.0368 

270 

72900 

19683  ooo 

16.4317 

6.4633 

221 

48841 

10  793  861 

14.8661 

6.0459 

271 

73441 

19  902  511 

16.4621 

6.4713 

222 

49284 

10  941  048 

14.8997 

6.0550 

272 

73984 

20  123  648 

16.4924 

6.4792 

223 

49729 

II  089  567 

14.9332 

6.0641 

273 

7  45  29 

20  346  417 

16.5227 

6  .  4872 

224 

501  76 

ii  239  424 

14.9666 

6.0732 

274 

75076 

20  570  824 

16.5529 

6.4951 

225 

50625 

n  390  625 

15.0000 

6.0822 

275 

75625 

20  796  875 

16.5831 

6  5030 

226 

51076 

ii  543  176 

15.0333 

6.0912 

276 

76176 

21  024  576 

16.6132 

6.5108 

227 

51529 

ii  697  083 

15.0665 

6.IOO2 

277 

76729 

21  253  933 

16.6433 

6.5187 

228 

51984 

II  852  352 

15.0997 

6.1091 

278 

77284 

21  484  952 

16.6733 

6.5265 

229 

52441 

12  008  989 

15.1327 

6.1180 

279 

77841 

21  717  639 

16.7033 

6.5343 

230 

52900 

12  167  OOO 

15.1658 

6.1269 

280 

78400 

21  952  OOO 

16-7332 

6.5421 

231 

53361 

12  326  391 

15.1987 

6.1358 

281 

78961 

22  188  041 

16  .  7631 

6.5499 

232 

53824 

12  487  168 

15.2315 

6.1446 

282 

7  95  24     22  425  768 

16.7929 

6.5577 

233 

54289 

12  649  337 

15.2643 

6.1534 

283 

8  oo  89 

22  665  IS? 

16.8226 

6.5654 

234 

54756 

12  8l2  904 

15.2971 

6.1622 

284 

80656 

22  906  304 

16.8523 

6.5731 

235 

55225 

12  977  875 

15.3297 

6.1710 

285 

8  12  25 

23  149  125 

16.8819 

6.5808 

236 

55696 

13  144  256 

15.3623 

6.1797 

286 

8  17  96 

23  393  656 

16.9115 

6.5885 

237 

56169 

13  312  053 

15.3948 

6.1885 

287 

8  23  69 

23  639  903 

16.9411 

6.5962 

238 

56644 

13  481  272 

15.4272 

6.1972 

288 

82944 

23  887  872 

16.9706 

6.6039 

239 

5  71  21 

13  651  919 

15.4596 

6.2058 

289 

83521 

24  137  569 

17.0000 

6.6115 

240 

576oo 

13  824  ooo 

15.4919 

6.2145 

290 

841  oo 

24  389  ooo 

17-0294 

6.6191 

241 

58081 

13  997  521 

15.5242 

6.2231 

291 

84681 

24  642  171 

17.0587 

6.6267 

242 

58564 

14  172  488 

15.5563 

6.2317 

292 

85264 

24  897  088 

17.0880 

6.6343 

243 

5  9049 

14  348  907 

15.5885 

6.2403 

293 

85849 

25  153  757 

17.1172 

6.6419 

244 

59536 

14  526  784 

15.6205 

6.2488 

294 

86436 

25  412  184 

17.1464 

6.6494 

245 

6  oo  25 

14  706  125 

15.6525 

6.2573 

295 

87025 

25  672  375 

17.1756 

6.6569 

246 

605  16 

14  886  936 

15-6844 

6.2658 

296 

87616 

25  934  336 

17.2047 

6.6644 

247 

6  10  09 

15  069  223 

15.7162 

6.2743 

297 

882  09 

26  198  073 

17.2337 

6.6719 

248 

6  15  04 

IS  252  992 

15.7480 

6.2828 

298 

88804 

26  463  592 

17.2627 

6.6794 

249 

6  2001 

15  438  249 

15-7797 

6.2912 

299 

89401 

26  730  899 

17.2916 

6.6869 

250 

6  25  oo 

15  625  ooo 

15.8114 

6.2996 

300 

90000 

27000000 

17-3205 

6.6943 

CHAIN  SURVEYING  65 

SQUARES,  CUBES,  SQUARE  ROOTS  AND  CUBE  ROOTS     (Continued) 


Nos. 

Squares. 

Cubes. 

Square 
root. 

Cube 
root. 

Nos. 

Squares. 

Cubes. 

Square 
root. 

Cube 
root. 

301 

9  (36  01 

27  270  901 

17.3494 

6.7018 

351 

12  32  oi 

43  243  551 

18-7350 

7.0540 

302 

91204 

27  543  608 

I7-378I 

6.7092 

352 

12  39  04 

43  614  208 

18.7617 

7.0607 

303 

91809 

27  818  127 

17.4069 

6.7166 

353 

124609 

43  986  977 

18.7883 

7.0674 

304 

9  24  16 

28  094  464 

17.4356 

6.7240 

354 

12  53  16 

44  361  864 

18.8149 

7.0740 

305 

93025 

28  372  625 

17.4642 

6.7313 

355 

12  6025 

44  738  875 

18.8414 

7.0807 

306 

93636 

28  652  616 

17.4929 

6.7387 

356 

12  67  36 

45  118  016 

18.8680 

7.0873 

307 

94249 

28  934  443 

17.5214 

6.7460 

357 

12  74  49 

45  499  293 

18.8944 

7.0940 

308 

94864 

29  2l8  112 

17  5499 

6.7533 

358 

12  81  64 

45  882  712 

18.9209 

7.1006 

309 

9548i 

29  503  629 

17.5784 

6.7606 

359 

12  88  81 

46  268  279 

18.9473 

7.1072 

310 

9  61  oo 

29  791  ooo 

17.6068 

6.7679 

360 

12  96  OO 

46  656  ooo 

18.9737 

7.II38 

3ii 

96721 

30  080  231 

17  6352 

6.7752 

36i 

13  03  21 

47  045  881 

19.0000 

7  .  1204 

312 

97344 

30  371  328 

17  6635 

6.7824 

362 

I3I044 

47  437  928 

19.0263 

7.1269 

3U 

97969 

30  664  297 

17.6918 

6.7897 

363 

I3I769 

47  832  147 

19  0526 

7-1335 

3i4 

985  96 

30  959  144 

17.7200 

6.7969 

364 

132496 

48  228  544 

19.0788 

7.1400 

315 

99225 

31  255  875 

17.7482 

6.8041 

365 

133225 

48  627  125 

19  1050 

7.1466 

3i6 

99856 

31  554  496 

17.7764 

6.8113 

366 

13  39  56 

49  027  896 

19.1311 

7  1531 

3i7 

10  04  89 

31  855  013 

17.8045 

6.8185 

367 

134689 

49  430  863 

19.1572 

7.1596 

318 

10  II  24 

32  157  432 

17  8326 

6.8256 

368 

13  54  24 

49  836  032 

19  183,3 

7.1661 

3i9 

10  17  61 

32  461  759 

17.8606 

6.8328 

369 

13  61  61 

50  243  409 

19.2094 

7.1726 

320 

1024  oo 

32  768000 

17.8885 

6.8399 

370 

136900 

50  653  ooo 

19-2354 

7-I79I 

321 

10  30  41 

33  076  161 

17.9165 

6.8470 

371 

13  76  41 

51  064  811 

19.2614 

7.1855 

322 

10  36  84 

33  .386  248 

17-9444 

6.8541 

372 

13  83  84 

Si  478  848 

19.2873 

7.1920 

323 

10  43  29 

33  698  267 

17.9722 

6.8612 

373 

13  9i  29 

51  895  117 

19.3132 

7.1984 

324 

10  49  76 

34  012  224 

18.0000 

6.8683 

374 

139876 

52  313  624 

19  3391 

7.2048 

325 

10  56  25 

34  328  125 

18.0278 

6.8753 

375 

140625 

52  734  375 

19  3649 

7.2112 

326 

10  62  76 

34  645  976 

18.0555 

6.8824 

376 

14  13  76 

53  157  376 

19.3907 

7-2177 

327 

10  69  29 

34  965  783 

18.0831 

6.8894 

377 

14  21  29 

53  582  633 

19.4165 

7.2240 

328 

10  75  84 

35  287  552 

18.1108 

6.8964 

378 

14  28  84 

54  oio  152 

19.4422 

7-2304 

329 

10  82  41 

35  611  289 

18.1384 

6.9034 

379 

14  36  41 

54  439  939 

19.4679 

7-2368 

330 

108900 

35  937  ooo 

18.1659 

6.9104 

38o 

144400 

54  872  ooo 

19.4936 

7.2432 

331 

10  95  61 

36  264  601 

18.1934 

6.9174 

38i 

14  Si  61 

55  306  341 

19.5192 

7.2495 

332 

II  02  24 

36  594  368 

I8.220Q 

6.9244 

382 

14  59  24 

55  742  968 

19-5448 

7  2558 

333 

II  0889 

36  926  037 

18.2483 

6.9313 

383 

56  181  887 

19-5704 

7  2622 

334 

ii  15  56 

37  259  704 

18.2757 

6.9382 

384 

14  74  56 

56  623  104 

19.5959 

7^2685 

335 

II  22  25 

37  595  375 

18.3030 

6-9451 

385 

14  82  25 

57  066  625 

19.6214 

7.2748 

336 

II  28  96 

37  933  056 

18.3303 

6.9521 

386 

148996 

57  512  456 

19.6469 

7.2811 

337 

ii  35  69 

38  272  753 

18.3576 

387 

149769 

57960603 

19.6723 

7.2874 

338 

II  42  44 

38  614  472 

18.3848 

6.9658 

388 

150544 

58  411  072 

19.6977 

7.2936 

339 

II  49  21 

38  958  219 

18.4120 

6.9727 

389 

15  13  21 

58  863  869 

19-7231 

7.2999 

340 

ii  56  oo 

39  304  ooo 

18.4391 

6.9795 

390 

IS  21  OO 

59  319  ooo 

19-7484 

7.3061 

341 

ii  62  81 

39  651  821 

18.4662 

6.9864 

391 

15  28  81 

59  776  471 

19-7737 

7.3124 

342 

ii  69  64 

40  ooi  688 

18.4932 

6.9932 

392 

15  36  64 

60  236  288 

19.7990 

7.3186 

343 

ii  76  49 

40  353  607 

18.5203 

7.0000 

393  15  44  49 

60  698  457 

19.8242 

7.3248 

344 

ii  83  36 

40  707  584 

18.5472 

7.0068 

394  15  52  36 

61  162  984 

19.8494 

7-3310 

345 

119025 

41  063  625 

18.5742 

7.0136 

395  15  60  25 

61  629  875 

19.8746 

7-3372 

346 

ii  97  16 

41  421  736 

18.6011 

7-0203 

396 

15  68  16 

62  099  136 

19.8997 

7-3434 

347 

12  04  09 

41  781  923 

18.6279 

7.0271 

397  i  15  76  09 

62  570  773 

18.9249 

7.3496 

348 

12  ii  04 

42  144  192 

18.6548 

7.0338 

398  15  84  04 

63  044  792 

19.9499 

7.3558 

349 

12  18  01 

42  508  549 

18.6815 

7.0406 

399  IS  92  oi 

63  521  199 

19.9750 

7.3619 

350 

12  25  OO 

42  875  ooo 

18.7083 

7-0473 

400 

16  oo  oo 

64  ooo  ooo 

20.0000 

7.3681 

66  PRACTICAL  SURVEYING 

SQUARES,  CUBES,  SQUARE  ROOTS  AND  CUBE  ROOTS     (Continued) 


Nos. 

Squares. 

Cubes. 

Square 
root. 

Cube 
root. 

Nos. 

Squares. 

Cubes. 

Square 
root. 

Cube 
root. 

401 

160801 

64  481  2OI 

20.0250 

7-3742 

45i 

20  34  01 

91  733  851 

21.2368;  7.6688 

402 

16  16  04 

64  964  808 

20.0499 

7.3803 

452 

20  43  04 

92  345  408 

21.2603  7-6744 

403 

162409 

65  450  827 

20.0749 

7.3864 

453 

20  52  09 

92  959  677 

21.2838  7.6801 

404 

16  32  16 

65  939  264 

20.0998 

7-3925 

454 

20  61  16 

93  576  664 

21.3073!  7-6857 

405 

16  40  25 

66  430  125 

20.1246 

7.3986 

455 

20  70  25 

94  196  375 

21.3307 

7.6914 

406 

16  48  36 

66  923  416 

20.1494 

7.4047 

456 

20  79  36 

94  818  816 

21.3542 

7.6970' 

407 

16  56  49 

67  419  143 

20.1742 

7.4108 

457 

20  88  49 

95  443  993 

21.3776;  7.7026 

408 

16  64  64 

67  917  312 

20.1990 

7-4169 

458 

20  97  64 

96  071  912 

21  .  4009 

7.7082 

409 

16  72  81 

68  417  929 

20.2237 

7-4229 

459 

21  0681 

96  702  579 

21.4243 

7.7138 

410 

1681  oo 

68  921  ooo 

20.2485 

7.4290 

460 

21  l6  OO 

97  336  ooo 

21.4476 

7.7194 

411 

16  89  21 

69  426  531 

20.2731 

7-4350 

461 

21  25  21 

97  972  181 

21.4709 

7.7250 

412 

16  97  44 

69  934  528 

20.2978 

7.4410 

462 

213444 

98  611  128 

21.4942 

7.7306 

4i3 

17  05  69 

70  444  997 

20.3224 

7.4470 

463 

21  43  69 

99  252  847 

21.5174  7.7362 

414 

I7I396 

70  957  944 

20.3470 

7-4530 

464 

21  5296 

99  897  344 

21-5407 

7-7418 

4iS 

17  22  25 

71  473  375 

20.3715 

7-4590 

465 

21  62  25 

loo  544  625 

21.5639 

7-7473 

416 

17  30  56 

71  991  296 

20.3961 

7-4650 

466 

21  71  56 

101  194  696 

21.5870 

7-7529 

417 

17  38  89 

72  511  713 

20.4206 

7-4710 

467 

21  8O  89 

1  01  847  563 

2I.6I02 

7.7584 

418 

17  47  24 

73  034  632 

20.4450 

7.4770 

468 

21  90  24 

102  503  232 

21.6333 

7.7639 

419 

17  55  61 

73  56o  059 

20.4695 

7.4829 

469 

21  99  61 

103  161  709 

21  .  6564 

7.7695 

420 

17  64  oo 

74  088  ooo 

20.4939 

7.4889 

470 

22*6900 

103  823  ooo 

21.6795 

7-7750 

421 

17  72  41 

74  618  461 

20.5183 

7-4948 

47i 

22  l8  41 

104  487  in 

21.7025 

7-7«05 

422 

17  80  84 

75  151  448 

20.5426 

7.5007 

472 

22  27  84 

105  154  048 

21  .  7256 

7.7860 

423 

17  89  29 

75  686  967 

20.5670 

7.5067 

473 

22  37  29 

105  823  817 

21.7486 

7.7915 

424 

17  97  76 

76  225  024 

20.5913 

7-5126 

474 

22  46  76 

106  490  424 

21.7715 

7-7Q70 

425 

18  06  25 

76  765  625 

20.6155 

7-5185 

475 

22  56  25 

107  171  875 

21-7945 

7.8025 

426 

18  14  76 

77  308  776 

20.6398 

7-5244 

476 

22  65  76 

107  850  176 

21.8174 

7.8079 

427 

18  23  29 

77  854  483 

20  .  6640 

7-5302 

477 

22  75  29 

1  08  531  333 

21  .  8403 

7.8134 

428 

18  31  84 

78  402  752 

20.6882 

7.536i 

478 

22  84  84 

109  215  352 

21.8632 

7.8188 

429 

18  40  41 

78  953  589 

20.7123 

7-5420 

479 

22  94  41 

109  902  239 

2I.886I 

7.8243 

430 

18  4900 

79  507  ooo 

20.7364 

7-5478 

480 

23  04  oo 

no  592  ooo 

21.9089 

7-8297 

43i 

18  57  61 

80  062  991 

20.7605 

7-5537 

481 

23  13  61 

III  284  641 

21.9317 

7.8352 

432 

18  66  24 

80  621  568 

20.7846 

7-5595 

482 

23  23  24 

111980168  21.9545 

7.8406 

433 

18  74  89 

81  182  737 

20.8087 

7.5654 

483 

23  32  89 

112678587  21.9773 

7.8460 

434 

18  83  56 

81  746  504 

20.8327 

7-5712 

484 

23  42  56 

H3  379  904'  22.0000 

7.8514 

435 

18  92  25 

82  312  875 

20.8567 

7-5770 

485 

23  52  25 

114  084  125 

22.0227 

7.8568 

436 

19  oo  96 

82  881  856 

20.8806 

7-5828 

486 

23  61  96 

114  791  256 

22.0454 

7.8622 

437 

19  09  69 

83  453  453 

20.9045 

7.5886 

487 

23  7i  69 

115  501  303 

22.0681 

7.8676 

438 

19  18  44 

84  027  672 

20.9284 

7-5944 

488 

23  81  44 

116  214  272 

22.0907 

7.8730 

439 

19  27  21 

84  604  519 

20.9523 

7.6001 

489 

23  91  21 

116  930  169 

22.1133 

7.8784 

440 

193600 

85  184  ooo 

20.9762 

7.6059 

490 

24  01  oo 

117  649  ooo 

22.1359 

7-8837 

441 

19  44  8l 

85  766  121 

2I.OOOO 

7.6117 

491 

24  10  81 

118  370  771 

22.1585 

7.8891 

442 

19  53  64 

86  350  888 

.21.0238 

7.6i74 

492 

24  20  64 

119  095  488 

22.I8II 

7.8944 

443 

19  62  49 

86  938  307 

21.0476 

7-6232 

493 

24  30  49 

119  823  157 

22.2036 

7.8998 

444 

19  71  36 

87  528  384 

21.0713 

7.6289 

494 

24  40  36 

120  553  784 

22.2261 

7.9051 

445 

19  80  25 

88  121  125 

21.0950 

7.6346 

495 

24  50  25 

121  287  375 

22  .  2486 

7.9105 

446 

19  89  16 

88  716  536 

2I.II87 

7.6403 

496 

2460  16 

122  023  936 

22.2711 

7.9158 

447 
448 

199809 
20  07  04 

89  314  623 
89  915  392 

21.1424 
21  .  1660 

7.6460 
7-6517 

497 
498 

24  70  09 
24  80  04 

122  763  473 
123  505  992 

22.2935 
22.3159 

7-92II 
7.9264 

449 

20  16  01 

90  518  849 

21  .  1896 

7-6574 

499 

24  90  01 

124  251  499!  22.3383 

7  9317 

450 

20  25  oo 

91  125  ooo 

21.2132 

7-6631 

500 

25  oo  oo 

125  ooo  ooo  22.3607 

1 

7-9370 

CHAIN  SURVEYING  67 

SQUARES,  CUBES,  SQUARE  ROOTS  AND  CUBE  ROOTS     (Continued) 


Nos. 

Squares. 

Cubes. 

Square 
root. 

Cube 
root. 

Nos. 

Squares. 

Cubes. 

Square 
root. 

Cube 
root. 

SGI 

25  10  01 

125  751  501 

22.3830 

7.9423 

551 

30  36  oi 

167  284  151 

23.4734 

8.1982 

502 

25  20  04 

126  506  008 

22.4054 

7.9476 

552 

30  47  04  168  196  608 

23-4947 

8.2031 

503 

25  30  09 

127  263  527 

22.4277 

7-9528 

553 

30  58  03 

169  112  377 

23.5160 

8.2081 

504 

25  40  16 

128  024  064 

22.4499 

7.958i 

554 

30  69  16 

170  031  464 

23.5372 

8.2130 

505 

25  So  25 

128  787  625 

22.4722 

7.9634 

555 

30  80  25 

170  953  875 

23.5584 

8.2180 

506 

25  60  36 

129  554  216 

22.4944 

7.9686 

556 

30  91  36 

171  879  616 

23.5797 

8.2229 

So? 

25  70  49 

130  323  843 

22.5167 

7-9739 

557 

31  02  49 

172  808  693 

23.6008 

8.2278 

508 

25  80  64 

131  096  512 

22.5389 

7-9791 

558 

3i  13  64 

173  741  H2 

23  .  6220 

8.2327 

509 

25  90  81 

131  872  229 

22.5610 

7-9843 

559 

3i  24  81 

174  676  879 

23.6432 

8.2377 

5io 

26  01  oo 

132  651  ooo 

22.5832 

7.9896 

56o 

313600 

175  616  ooo 

23.6643 

8.2426 

Sii 

26  II  21 

133  432  831 

22.6053 

7.9948 

56l 

31  47  21 

176  558  481 

23.6854 

8.2475 

512 

26  21  44 

134  217  728 

22.6274 

8.0000 

562 

31  58  44 

177  504  328 

23.7065 

8.2524 

513 

26  31  69 

135  005  697 

22.6495 

8.0052 

563 

31  69  69 

178  453  547 

23.7276 

8.2573 

514 

26  41  96 

135  796  744 

22.6716 

8.0104 

564 

318096 

179  406  144 

23-7487 

8.2621 

515 

26  52  25 

136  590  875 

22.6936 

8.0156 

565 

31  92  25 

180  362  125 

23.7697 

8.2670 

Si6 

26  62  56 

137  388  096 

22.7156 

8.0208 

566 

32  03  56 

181  321  496 

23.7908 

8.2719 

Si? 

26  72  89 

138  188  413 

22.7376 

8.0260 

567 

32  14  89 

182  284  263 

23.8118 

8.2768 

Si8 

26  83  24 

138  991  832 

22.7596 

8.0311 

568 

32  26  24 

183  250  432 

23.8328 

8.2816 

519 

26  93  61 

139  798  359 

22.7816 

8.0363 

569 

32  37  61 

184  220  009 

23.8537 

8.2865 

520 

27  04  oo 

140  608  ooo 

22.8035 

8.0415 

570 

324900 

185  193  ooo 

23-8747 

8.2913 

521 

27  14  41 

141  420  761 

22.8254 

8.0466 

571 

326041 

186  169  411 

23.8956 

8.2962 

522 

27  24  84 

142  236  648 

22.8473 

8.0517 

572 

32  71  84 

187  149  248 

23.9165 

8.3010 

523 

27  35  29 

143  055  667 

22.8692 

8.0569 

573 

32  83  29 

188  132  517 

23-9374 

8.3059 

524 

27  45  76 

143  877  824 

22.8910 

8.0620 

574 

32  94  76 

189  119  224 

23.9583 

8.3107 

525 

27  56  25 

144  703  125 

22.9129 

8.0671 

575 

330625 

190  109  375 

23.9792 

8.3155 

526 

27  66  76 

145  531  576 

22.9347 

8.0723 

576 

33*776 

191  102  976 

24.0000 

8.3203 

527 

27  77  29 

146  363  183 

22.9565 

8.0774 

577 

33  29  29 

192  100  033 

24.0208 

8-3251 

528 

27  8?  84 

147  197  952 

22.9783 

8.0825 

578 

334084 

193  100  552 

24.0416 

8.3300 

529 

27  98  41 

148  035  889 

23.0000 

8.0876 

579 

33  52  41 

194  104  539 

24  .  0624 

8-3348 

530 

280900 

148  877  ooo 

23.0217 

8.0927 

58o 

33  64  oo 

195  l\2  OOO 

24.0832 

8.3396 

531 

28  19  61 

149  721  291 

23  0434 

8.0978 

58l 

33  75  61 

196  122  941 

24.1039 

8.3443 

532 

28  30  24 

150  568  768 

23.0651 

8.1028 

582 

338724 

197  137  368 

24.1247 

8-3491 

533 

28  40  89 

151  419  437 

23.0868 

8.1079 

583 

33  98  89 

198  155  287 

24.1454 

8.3539 

534 

28  51  56 

152  273  304 

23.1084 

8.1130 

584 

34  10  56 

199  176  704 

24.1661 

8.3587 

535 

28  62  25 

153  130  375 

23.1301 

8.1180 

585 

34  22  25 

2OO  2OI  625 

24.1868 

8.3634 

536 

287296 

153  990  656 

23.1517 

8.1231 

586 

343396 

201  230  056 

24.2074 

8.3682 

537 

288369 

154  854  153 

23-1733 

8.1281 

587 

34  45  69 

202  262  003 

24.2281 

8.3730 

538 

28  94  44 

155  720  872 

23.1948 

8.1332 

588 

34  57  44 

203  297  472 

24.2487 

8.3777 

539 

29  05  21 

156  5QO  SlQ 

23.2164 

8.1382 

589 

34  69  21 

204  336  469 

24.2693 

8.3825 

540 

29  1600 

157  464  ooo 

23.2379 

8.1433 

590 

348100 

205  379  ooo 

24.2899 

8.3872 

541 

29  26  81 

158  340  421 

23.2594 

8.1483 

591 

34  92  8l 

206  425  071 

24.3105 

8.3919 

542 

29  37  64 

159  220  088 

23.2809 

8.1533 

592 

35  04  64 

207  474  688 

24-3311 

8.3967 

543 

29  48  49 

160  103  007 

23.3024 

8.1583 

593 

35  16  49 

208  527  857 

24.3516 

8.4014 

544 

29  59  36 

160  989  184 

23-3238 

8.1633 

594 

35  28  36 

209  584  584 

24.3721 

8.4061 

545 

29  70  25 

161  878  625 

23.3452 

8.1683 

595 

35  40  25 

210  644  8-5 

24-.  3926 

8.4108 

546 

29  81  16 

162  771  336 

23.3666 

8.1733 

596 

35  52  16 

211  708  736 

24.4131 

8.4155" 

547 

2992  09 

163  667  323 

23.3880 

8.1783 

597 

35  6409 

212  776  173 

24.4336 

8.4202^ 

548 

30  03  04 

164  566  592 

23.4094 

8.1833 

598 

35  76  04 

213  847  192 

24.4540 

8.4249 

549 

30  14  oi 

165  469  I4Q 

23.4307 

8.1882 

599 

35  88  oi 

214  921  799 

24.4745 

8.4296 

550 

302500 

166  375  ooo 

23  4521 

8.1932 

600 

36  oo  oo 

2l5  OOO  OOO 

24.4949 

8.4343 

68  PRACTICAL  SURVEYING 

SQUARES,  CUBES,  SQUARE  ROOTS  AND  CUBE  ROOTS     (Continued) 


Nos. 

Squares. 

Cubes. 

Square 
root. 

Cube 

root. 

Nos. 

Squares. 

Cubes. 

Square 
root. 

Cube 
root. 

601 

36  12  01 

217  081  801 

24.5153 

8.4390 

651 

42  38  01 

275  894  451 

25.5147 

8.6668 

602 

36  24  04 

218  167  208 

24-5357 

8.4437 

652 

42  51  04 

277  167  808 

25.5343 

8.6713 

603 

36  36  09 

219  256  227 

24.5561 

8.4484 

653 

42  64  09 

278  445  077 

25  5539 

8.6757 

604 

36  48  16 

220  348  864 

24-5764 

8.4530 

654 

42  77  16 

279  726  264 

25.5734 

8.6801 

605 

36  60  25 

221  445  125 

24.5967 

8.4577 

655 

429025 

281  oil  375 

25  5930 

8.6845 

606 

367236 

222  545  016 

24.6171 

8.4623 

656 

43  03  36 

282  300  416 

25.6125 

8.6890 

607 

36  84  49 

223  648  543 

24-6374 

8.4670 

657 

43  16  49 

283  593  393 

25.6320 

8.6934 

608 

369664 

224  755  712 

24.6577 

8.4716 

658 

43  29  64 

284  890  312 

25  6515  8.6978 

609 

37  08  81 

225  866  529 

24.6779 

8.4763 

659 

43  42  81 

286  191  179 

25.6710!  8.7022 

610 

372100 

226  981  ooo 

24-6982 

8.4809 

660 

435600 

287  496  ooo 

25.6905 

8.7066 

611 

37  33  21 

228  099  131 

24.7184 

8.4856 

661 

43  69  21 

288  804  781 

25.7099 

8.7110 

612 

37  45  44 

229  220  928 

24.7386 

8.4902 

662 

43  82  44 

290  117  528 

25.7294 

8.7154 

613 

37  57  69 

230  346  397 

24.7588 

8.4948 

663 

43  95  69 

291  434  247 

25.7488 

8.7198 

614 

376996 

231  475  544 

24.7790 

8.4994 

664 

440896 

292  754  944 

25.7682 

8.7241 

615 

37  82  25 

232  608  375 

24.7992 

8.5040 

665 

44  22  25 

294  079  625 

25.7876 

8.7285 

616 

37  94  56 

333  744  896 

24.8193 

8.5086 

666 

44  35  56 

295  408  296 

25  8070 

8-7329 

617 

380689 

234885  113 

24.8395 

8.5132 

667 

444889 

296  740  963 

25  .  8263 

8.7373 

618 

38  19  24 

236  029  032 

24.8596 

8.5178 

668 

44  62  24 

298  077  632 

25  8457 

8.7416 

619 

38  31  61 

237  176  659 

24.8797 

8.5224 

669 

44  75  6l 

299  418  309 

25  .  8650 

8.7460 

620 

38  44  oo 

238  328  ooo 

24.8998 

8.5270 

670 

448900 

300  763  ooo 

25.8844 

8.7503 

621 

38  56  41 

239  483  061 

24.9199 

8.5316 

671 

45  02  41 

302  III  711 

25.9037 

8.7547 

622 

386884 

240  641  848 

24-9399 

8.5362 

672 

45  15  84 

303  464  448 

25.9230 

8-7590 

623 

38  81  29 

241  804  367 

24.9600 

8.5408 

673 

45  29  29 

304  821  217 

25.9422 

8.7634 

624 

38  93  76 

242  970  624 

24.9800 

8.5453 

674 

45  42  76 

306  182  024 

25.9615 

8.7677 

625 

390625 

244  140  625 

25.0000 

8.5499 

675 

45  56  25 

307  546  875 

25.9808 

8.7721 

626 

39  18  76 

245  314  376 

25  .  02OO 

8.5544 

676 

456976 

308  915  776 

26.0000 

8.7764 

627 

39  3i  29 

246  491  883 

25.0400 

8.5590 

677 

45  83  29 

310  288  733 

26.0192 

8.7807 

628 

39  43  84 

247  673  152 

25-0599 

8  5635 

678 

459684 

311  665  752 

26.0384 

8.7850 

629 

39  56  41 

248  858  189 

25.0799 

8.5681 

679 

46  10  41 

313  046  839 

26.0576 

8.7893 

630 

396900 

250  047  ooo 

25.0998 

8.5726 

680 

462400 

314  432  ooo 

26.0768 

8-7937 

631 

39  81  61 

251  239  59i 

25-1197 

8.5772 

681 

46  37  6l 

315  821  241 

26.0960 

8.7980 

632 

39  94  24 

252  435  968 

25.1396 

8.5817 

682 

46  51  24 

317  214  568 

26.1151 

8.8023 

633 

400689 

253  636  137 

25.1595 

8.5862 

683 

46  64  89 

318  611  987 

26.1343 

8.8066 

634 

40  19  56 

254  840  104 

25.1794 

8.5907 

684 

46  78  56 

320  013  504 

26.1534 

8.8109 

635 

40  32  25 

256  047  875 

25.1992 

8.5952 

685 

46  92  25 

321  419  125 

26.1725 

8.8152 

636 

404496 

257  259  456 

25.2190 

8.5997 

686 

47  05  96 

322  828  856 

26.1916 

8.8194 

637 

40  57  69 

258  474  853 

25.2389 

8.6043 

687 

47  19  69 

324  242  703 

26.2107 

8.8237 

638 

40  70  44 

259  694  07- 

25.2587 

8.6088 

688 

47  33  44 

325  660  672 

26  .  2298 

8.8280 

639 

40  83  21 

260  917  119 

25.2784 

8.6132 

689 

47  47  21 

327  082  769 

26.2488 

8.8323 

640 

40  96  oo 

262  144  ooo 

25.2982 

8.6177 

690 

47  61  oo 

328  509  ooo 

26.2679 

8.8366 

641 

41  08  81 

263  374  721 

25.3180 

8.6222 

691 

47  74  81 

329  939  371 

26.2869 

8.8408 

642 

41  21  64 

264  609  288 

25  3377 

8.6267 

692 

47  88  64 

331  373  888 

26.3059 

8.8451 

643 

41  34  49 

265  847  707 

25-3574 

8.6312 

693 

48  02  49 

332  812  557 

26.3249 

8.8493 

644 

41  47  36 

267  089  984 

25-3772 

8.6357 

694 

48  16  36 

334  255  384 

26.3439 

8.8536 

645 

416025 

268  336  125 

25.3969 

8.6401 

695 

48  30  25 

335  702  375 

26  .  3629 

8.8578 

646 

41  73  16 

269  586  136 

25.4165 

8.6446 

696 

48  44  16 

337  153  536 

26.3818 

8.8621 

647 

41  86  09 

270  840  023 

25.4362 

8.6490 

697 

485809 

338  608  873 

26.4008 

8.8663 

648 

41  99  04 

272  097  792 

25.4558 

8.6535 

698 

48  72  04 

340  068  392 

26.4197 

8.8706 

649 

42  12  OI 

273  359  449 

25.4755 

8.6579 

699 

48  86  01 

341  532  099 

26.4386 

8.8748 

650 

4225  oc 

274  625  ooo 

25.4951 

8.6624 

700 

49  oo  oo 

343000000 

26.4575 

8.8790 

CHAIN   SURVEYING  69 

SQUARES,  CUBES,  SQUARE  ROOTS  AND  CUBE  ROOTS     (Continued) 


Nos. 

Squares. 

Cubes. 

Square 
root. 

Cube 
root. 

Nos. 

Squares. 

Cubes. 

Square 
root. 

Cube 
root. 

701 

49  14  01 

344  472  101 

26.4764 

8.8833 

751 

56  40  01 

423  564  75i 

27.4044 

9.0896 

702 

49  28  04 

345  948  408 

26.4053 

8.8875 

752 

56  55  04 

425  259  008 

27  .  4226 

9-0937 

703 

49  42  09 

347  428  927 

26.5141 

8.8917 

753 

56  7009 

426  957  777 

27.4408 

9-0977 

704 

49  56  16 

348  913  664 

26.5330 

8.8959 

754 

56  85  16 

428  66l  064 

27.4591 

9.1017 

70S 

49  70  25 

350  402  625 

26.5518 

8.9001 

755 

570025 

430  368  875 

27-4773 

9  1057 

706 

49  84  36 

351  895  816 

26.5707 

8.9043 

756 

57  15  36 

432  081  216 

27-4955 

9.1098 

707 

49  98  49 

353  393  243 

26  .  5895 

8.9085 

757 

57  30  49 

433  798  093 

27.5136 

9  H38 

708 

50  12  64 

354  894  912 

26  .  6083 

8.9127 

758 

57  45  64 

435  519  512 

27.5318 

9.1178 

709 

50  26  81 

356  400  829 

26.6271 

8.9169 

759 

57  6081 

437  245  479 

27.5500 

9.1218 

710 

50  41  oo 

3579HOOO 

26.6458 

8.9211 

760 

57  76oo 

438  976  ooo 

27.5681 

9  1258 

711 

50  55  21 

359  425  43i 

26.6646 

8.9253 

?6l 

57  91  21 

440711  081 

27.5862 

9.1298 

712 

50  69  44 

360  944  128 

26.6833 

8.9295 

762 

580644 

442  450  728 

27.6043 

9  1338 

713 

50  83  69 

362  467  097 

26.7021 

8.9337 

763 

58  21  69 

444  194  947 

27.6225 

9-1378 

7U 

50  97  96 

363  994  344 

26.7208 

8.9378 

764 

58  36  96 

445  943  744 

27  .  6405 

9.1418 

715 

51  12  25 

365  525  875 

26.7395 

8.9420 

765 

58  52  25 

447  697  125 

27.6586 

9  1458 

716 

51  26  56 

367  061  696 

26.7582 

8.9462 

766 

58  67  56 

449  455  096 

27.6767 

9.1498 

717 

51  40  89 

368  601  813 

26.7769 

8.9503 

767 

58  82  89 

451  217  663 

27.6948 

9-1537 

718 

Si  55  24 

370  146  232 

26.7955 

8.9545 

768 

58  98  24 

452  984  832 

27.7128 

9-1577 

719 

51  69  61 

371  694  959 

26.8142 

8.9587 

769 

59  13  61 

454  756  609 

27.7308 

9.1617 

720 

518400 

373  248  ooo 

26.8328 

8.9628 

770 

592900 

456  533  ooo 

27.7489 

9.1657 

721 

51  98  41 

374  805  361 

26.8514 

8.9670 

771 

59  44  4i 

458  314  on 

27.7669 

9.1696 

722 

52  12  84 

376  367  048 

26.8701 

8.9711 

772 

59  59  84 

460  099  648 

27.7849 

9.I736 

723 

52  27  29 

377  933  067 

26.8887 

8.9752 

773 

59  75  29 

461  889  917 

27.8029 

9-1775 

724 

52  41  76 

379  503  424 

26.9072 

8.9794 

774 

599076 

463  684  824 

27.8209 

9.I8I5 

'25 

52  56  25 

381  078  125 

26.9258 

8.9835 

775 

600625 

465  484  375 

27.8388 

9.1855 

726 

52  70  76 

382  657  176 

26.9444 

-8.9876 

776 

6021  76 

467  288  576 

27.8568 

9-1894 

727 

52  85  29 

384  240  583 

26.9629 

8.9918 

777 

6037  29 

469  097  433 

27.8747 

9  1933 

728 

529984 

385  828  352 

26.9815 

8.9959 

778 

605284 

470  910  952 

27.8927 

9-1973 

729 

53  14  4i 

387  420  489 

27.0000 

9.0000 

779 

6068  41 

472  729  139 

27.9106 

9.2012 

730 

532900 

389  017  ooo 

27.0185 

9.0041 

780 

608400 

474  552  ooo 

27.9285 

9.2052 

731 

53  43  61 

390  617  891 

27.0370 

9.0082 

78l 

609961 

476  379  541 

27.9464 

9.2091 

732 

53  58  24 

392  223  168 

27.0555 

9  0123 

782 

61  15  24 

478  211  768 

27.9643 

9  2130 

733 

53  72  89 

393  832  837 

27.0740 

9.0164 

783 

61  30  89 

480  048  687 

27.9821 

9.2170 

734 

53  87  56 

395  446  904 

27.0924 

9.0205 

784 

61  46  56 

481  890  304 

28.0000 

9.2209 

735 

54  02  25 

397  065  375 

27.1109 

9.0246 

785 

61  62  25 

483  736  625 

28.0179 

9.2248 

736 

541696 

398  688  256 

27.1293 

9.0287 

786 

61  77  96 

485  587  656 

28.0357 

9.2287 

737 

54  31  69 

400  315  553 

27-1477 

9.0328 

787 

61  93  69 

487  443  403 

28.0535 

9.2326 

738 

54  46  44 

401  947  272 

27.1662 

9-0369 

788 

620944 

489  303  872 

28.0713 

9-2365 

739 

54  61  21 

403  583  419 

27.1846 

9.0410 

789 

62  25  21 

491  169  069 

28.0891 

9.2404 

740 

5476oo 

405  224  ooo 

27.2029 

9.0450 

790 

62  41  oo 

493  039  ooo 

28.1069 

9-2443 

741 

54  90  8l 

406  869  021 

27.2213 

9.0491 

791 

62  56  81 

494  913  671 

28.1247 

9.2482 

742 

55  05  64 

408  518  488 

27.2397 

9.0532 

792 

62  72  64 

496  793  088 

28.1425 

9.2521 

743 

55  20  49 

410  172  407 

27  .  2580 

9.0572 

793 

6288  49 

498  677  257 

28.1603 

9-2560 

744 

55  35  36 

411  830  784 

27.2764 

9-0613 

794 

63  04  36 

500  566  184 

28.1780 

9-2599 

745 

55  50  25 

413  493  625 

27.2947 

9-0654 

795 

63  20  25 

502  459  875 

28.1957 

9-2638 

746 

55  65  16 

415  160  936 

27.3130 

9.0694 

796 

63  36  16 

504  358  336 

28.2135 

9.2677 

747 

558-009 

416  832  723 

27.3313 

9-0735 

797 

63  52  09 

506  261  573 

28.2312 

9.2716 

748 

55  95  04 

418  508  992 

27.3496 

9-0775 

798 

63  68  04 

508  169  592 

28.2489 

9-2754 

749 

56  10  01 

420  189  749 

27.3679 

9.0816 

799 

63  84  01 

510  082  399 

28.2666 

9-2793 

750 

562500 

421  875  ooo 

27.3861 

9.0856 

800 

64  oo  oo 

512  ooo  ooo 

28.2843 

9.2832 

70  PRACTICAL  SURVEYING 

SQUARES,  CUBES,  SQUARE  ROOTS  AND  CUBE  ROOTS     (Continued) 


Nos. 

Squares. 

Cubes. 

Square 
root. 

Cube 
root. 

Nos. 

Squares. 

Cubes. 

Square 
root. 

Cube 
root. 

801 

64  16  01 

513  922  401 

28.3019 

9  .  2870 

851 

72  42  01 

616  295  051 

29.1719 

9.4764 

802 

643204 

515  849  608 

28.3196 

9-2909 

852 

72  59  04 

618  470  208 

29.1890 

9.4801 

803 

64  48  09 

517  781  627 

28.3373 

9.2948 

853 

72  76  09 

620  650  477 

29  .  2062 

9.4838 

804 

64  64  16 

519  7i8  464 

28.3549 

9.2986 

854 

72  93  16 

622  835  864 

29.2233 

9.4875 

805 

64  80  25 

521  660  125 

28.3725 

9  3025 

855 

73  10  25 

625  026  375 

29.2404 

9.4912 

806 

649636 

523  606  616 

28.3901 

9.3063 

856 

73  27  36 

627  222  Ol6 

29-2575 

9.4949 

807 

65  12  49 

525  557  943 

28.4077 

9-3102 

857 

73  44  49 

629  422  793 

29.2746 

9-4986 

808 

65  28  64 

527514112 

28.4253 

9  3140 

858 

73  61  64 

631  628  712 

29.2916 

9.5023 

809 

65  44  81 

529  475  129 

28.4429 

9.3179 

859 

73  78  81 

633  839  779 

29.3087 

9.5060 

810 

65  61  oo 

531  441  ooo 

28.4605 

9.3217 

860 

7396oo 

636  056  ooo 

29.3258 

9-5097 

811 

65  77  21 

533  4ii  731 

28.4781 

9  3255 

861 

74  13  21 

638  277  38i 

29.3428 

9-5134 

812 

65  93  44 

535  387  328 

28.4956 

9.3294 

862 

74  30  44 

640  503  928 

29.3598 

9.5I7I 

813 

66  09  69 

537  367  797 

28.5132 

9-3332 

863 

74  47  69 

642  735  647 

29.3769 

9.5207 

814 

662596 

539  353  144 

28.5307 

9-3370 

864 

74  64  96 

644  972  544 

29-3939 

9-5244 

815 

66  42  25 

541  343  375 

28.5482 

9.3408 

865 

74  82  25 

647  214  625 

29.4109 

9-5281 

816 

66  58  56 

543  338  496 

28.5657 

9-3447 

866 

74  99  56 

649  461  896 

29.4279 

9-5317 

817 

66  74  89 

545  338  513 

28.5832 

9.3485 

867 

75  16  89 

651  714  363 

29.4449 

9-5354 

818 

66  91  24 

547  343  432 

28.6007 

9-3523 

868 

75  34  24 

653  972  032 

29.4618 

9.5391 

819 

67  07  61 

549  353  259 

28.6182 

9.356I 

869 

75  51  61 

656  234  909 

29.4788 

9-5427 

820 

67  24  oo 

551  368  ooo 

28.6356 

9-3599 

870 

75  6900 

658503000  29.4958 

9.5464 

821 

67  40  41 

553  387  601 

28.6531 

9.3637 

871 

75  86  41 

660  776  311 

29.5127 

9.5501 

822 

67  56  84 

555  412  248 

28  .  6705 

9.3675 

872 

76  03  84 

663  054  848 

29.5296 

9-5537 

823 

67  73  29 

557  441  767 

28.6880 

9.3713 

873 

76  21  29 

665  338  617 

29.5466 

9-5574 

824 

67  89  76 

559  476  224 

28.7054 

9-3751 

874 

76  38  76 

667  627  624 

29.5635 

9.5610 

825 

68  06  25 

561  515  625 

28.7228 

9.3789 

875 

76  56  25 

669  921  875 

29.5804 

9.5647 

826 

68  22  76 

563  559  976 

28.7402 

9-3827 

876 

76  73  ?6 

672  221  376 

29.5973 

9-568.3 

827 

68  39  29 

565  609  283 

28.7576 

9.3865 

877 

76  91  29 

674  526  133 

29.6142 

9.5719 

828 

68  55  84 

567  663  552 

28.7750 

9.3902 

878 

77  08  84 

676  836  152 

29-6311 

9.5756 

829 

68  72  41 

569  722  789 

28.79^4 

9-3940 

879 

77  26  41 

679  151  439 

29.6479 

9-5792 

830 

688900 

571  787  ooo 

28.8097 

9.3978 

880 

77  4400 

68  1  472  ooo 

29.66*8 

9.5828 

831 

69  05  61 

573  856  191 

28.8271 

9.4016 

881 

77  61  61 

683  797  841 

29.6816 

9.5865 

1832 

69  22  24 

575  930  368 

28.8444!  9.4053 

882 

77  79  24 

686  128  968 

29-6985 

9-5901 

i  833 

69  38  89 

578  009  537 

28.8617 

9.4091 

883 

77  96  89 

688  465  387 

29-7153 

9-5937 

834 

69  55  56 

580  093  704 

28.8791 

9.4129 

884 

78  14  56 

690  807  104 

29.7321 

9-5973 

,835 

69  72  25 

582  182  875 

28.8964 

9.4166 

885 

78  32  25 

693  154  125 

29.7489 

9.6010 

836 

69  88  96 

584  277  056 

28.9137 

9.4204 

886 

784996 

695  506  456 

29.7658 

9.6046 

837 

70  05  69 

586  376  253 

28.9310 

9.4241 

887 

78  67  69 

697  864  103 

29.7825 

9.6082 

838 
839 

70  22  44 
70  39  21 

588  480  472 
590  589  719 

28.9482 
28.9655 

9-4279 
9.4316 

888 
889 

78  85  44 
79  03  21 

700  227  072 
702  595  369 

29-7993 
29.8161 

9.6118 
9-6154 

840 

70  56  oo 

592  704  ooo 

28.9828 

9-4354 

890 

792100 

704  969  ooo 

29.8329 

9.6190 

841 

70  72  81 

594  823  321 

29.0000 

9.4391 

891 

79  38  81 

707  347  97i 

29  .  8496 

9.6226 

842 

70  89  64 

596  947  688 

29.0172 

9.4429 

892 

79  56  64 

709  732  288 

29.8664 

9.6262 

843 

71  06  49 

599  077  107 

29.0345 

9.4466 

893 

79  74  49 

712  121  957 

29.8831 

9.6298 

844 

71  23  36 

601  211  584 

29.0517 

9  -  4503 

894 

79  92  36 

714  5i6  984 

29.8998 

9-6334 

845 

71  40  25 

603  351  125 

29.0689 

9-4541 

895 

80  10  25 

716  917  375 

29.9166 

9.6370 

846 

71  57  16 

605  495  736 

29.0861 

9.4578 

896 

80  2g  16 

719  323  136 

29-9333 

9.6406 

847 

71  7409 

607  645  423 

29.1033 

9.4615 

897 

80  46  09 

721  734  273 

29.9500 

9.6442 

848 

71  91  04 

609  800  192 

29.1204 

9.4652 

898 

80  64  04 

724  150  792 

29.9666 

9-6477 

849 

72  08  01 

6n  960  049 

29.1376 

9.4690 

899 

80  82  or 

726  572  699 

29.9833 

9.6513 

850 

722500 

614  125  ooo 

29.1548 

9-4727 

900 

81  oo  oo 

729  ooo  ooo 

30.0000 

9  6549 

CHAIN  SURVEYING  71 

SQUARES,  CUBES,  SQUARE  ROOTS  AND  CUBE  ROOTS     (Continued) 


Nos. 

Squares. 

Cubes. 

Square 
root. 

Cube 
root. 

Nos. 

Squares. 

Cubes. 

Square 
root. 

Cube 
root. 

901 

81  18  01 

73i  432  701 

30.0167 

9.6585 

95: 

904401 

860  085  351 

30.8383 

9  8339 

902 

81  36  04 

733  870  808 

30.0333  9.6620 

952 

906304 

862  801  408 

30.8545 

9.8374 

903 

81  5409 

736  314  327 

30.0500!  9.6656 

953 

9082  09 

865  523  :77 

30.8707 

9.8408 

904 

81  72  16 

738  763  264 

30.0666 

9.6692 

954 

91  01  16 

868  250  664 

30.8869 

9.8443 

905 

81  9025 

741  217  625 

30.0832 

9.6727 

955 

91  20  25 

870  983  875 

30.903: 

9.8477 

906 

820836 

743  677  4i6 

30.0998 

9-6763 

956 

9:  39  36 

873  722  816 

30.9:92 

9-85II 

90? 

82  26  49 

746  142  643 

30.1164  9.6799 

957 

91  58  49 

876  467  493 

30.9354 

9.8546 

908 

82  44  64 

748  613  312 

30.1330  9-6834 

958 

91  77  64 

879  217  9:2 

30.95:6 

9.8580 

909 

82  62  81 

751  089  429 

30.1496 

9.6870 

959 

91  9681 

88  1  974  079 

30.9677 

9.8614 

910 

82  81  oo 

753  571  ooo 

30.1662 

9-690S 

960 

92  16  oo 

884  736  ooo 

30.9839 

9.8648 

911 

82  99  21 

756  058  031 

30.1828 

9.694: 

96! 

92  35  21 

887  503  681 

3:.  oooo 

9.8683 

912 

83  17  44 

758  550  528 

30.1993 

9.6976 

962 

925444 

890  277  128 

31.0161 

9-87:7 

83  35  69 

761  048  497 

30.2159 

9.7012 

963 

92  73  69 

893  056  347 

3:  0322 

9.875: 

914 

835396 

763  551  944 

30.2324 

9-7047 

964 

929296 

895  841  344 

3:.  0483 

9.8785 

83  72  25 

766  060  875 

30.2490 

9.7082 

965 

93  :2  25 

898  632  125 

3:.  0644 

9.8819 

916 

839056 

768  575  296 

30.2655 

9.7118 

966 

93  31  56 

901  428  696 

3:.  0805 

9-8854 

917 

840889 

771  095  213 

30.2820 

9.7:53 

967 

93  50  89 

904  231  063 

3:.  0966 

9.8888 

918 

84  27  24 

773  620  632 

30.2985 

9-7:88 

968 

93  70  24 

907  039  232 

3:.  "27 

9.8922 

919 

84  45  6l 

776  151  559 

30.3150 

9.7224 

969 

93  89  61 

909  853  209 

31.1288 

8.8956 

920 

8464  oo 

778  688  ooo 

30.3315 

9-7259 

970 

940900 

912  673  ooo 

3:.  1448 

9.8990 

921 

84  82  41 

781  229  961 

30.3480 

9.7294 

971 

94  28  4: 

915498611 

3:.:6o9 

9.9024 

922 

850084 

783  777  448 

30.3645 

9-7329 

972 

94  47  84 

918  330  048 

3:.:  769 

9.9058 

923 

85  19  29 

786  330  467 

30.3809 

9.7364 

973 

94  67  29 

921  167  317 

31.1929 

9.9092 

924 

85  37  76 

788  889  024 

30.3974 

0.7400 

974 

94  86  76 

924  oio  424 

31.2090 

9.9126 

925 

85  S6  25 

791  453  125 

30.4138 

9-7435 

975 

950625 

926  859  375 

3:-  2250 

9.9160 

926 

85  74  76 

794  022  776 

30.4302 

9.7470 

976 

95  25  76 

929  7:4  :?6 

3:-  24:0 

9.9:94 

927 

85  93  29 

796  597  983 

30.4467 

9.7505 

977 

95  45  29 

932  574  833 

3:.  2570 

9.9227 

928 

86  II  84 

799  178  752 

30.4631 

9-7540 

978 

95  64  84 

935  44:  352 

3:-  2730 

9.9261 

929 

86  30  41 

801  765  089 

30.4795 

9-7575 

979 

95  84  41 

938  3:3  739 

31.2890 

9.9295 

•930 

864900 

804  357  ooo 

30.4959 

9.7610 

980 

96  04  oo 

941  192  ooo 

31.3050 

9.9329 

931 

86  67  61 

806  954  491 

30.5:23 

9.7645 

98l 

962361 

944  076  141 

31.3209 

9.9363 

932 

86  86  24 

809  557  568 

30.5287 

9.7680 

982 

9643  24 

946  966  168 

9-9396 

933 

87  04  89 

812  166  237 

30.5450 

9.77:5 

983 

96  62  89 

949  862  087 

31.3528 

9-9430 

934 

87  23  56 

814  780  504 

30.5614 

9-7750 

?s- 

968256 

952  763  904 

31  .3688 

9.9464 

935 

87  42  25 

817  400  375 

30.5778 

9.7785 

97  02  25 

955  671  625 

3:  '3847 

9-9497 

936 

876096 

820  025  856 

30.5941 

9.7819 

986 

972196 

958  585  256 

31.4006 

9-9531 

937 

87  79  69 

822  656  953 

30.6105 

9.7854 

987 

974169 

961  504  803 

3:.  4:66 

9.9565 

938 

8?  98  44 

825  293  672 

30.6268 

9.7889 

988 

97  61  44  964  430  272 

3:  4325 

9  9598 

939 

88  17  21 

827  936  019 

30.6431 

9.7924 

989 

97  81  21 

967  361  669 

3:.  4484 

940 

88.3600 

830  584  ooo 

30.6594 

9-7959 

990 

9801  oo 

970  299  ooo 

3:-  4643 

9.9666 

941 

885481 

833  237  621 

30.6757 

9-7993 

991 

98  2081 

973  242  271 

31.4802 

9.9699 

942 

88  73  64 

835  896  888 

30.6920 

9.8028 

992 

984064 

976  191  488 

3:.  496o 

9-9733 

943 

889249 

838  561  807 

30.7083 

9.8063 

993 

986049 

979  146  657 

3:.5"9 

9.9766 

944 

89  II  36 

841  232  384 

30.7246 

9.8097 

994 

988036 

982  107  784 

3:-  5278 

9.9800 

945 

89  30  25 

843  908  625 

30.7409 

9-8132 

995 

990025 

985  074  875 

3:-  5436 

9.9833 

946 

89  49  16 

846  590  536 

30.757: 

9.8167 

996 

99  20  16 

988  047  936 

3:-  5595 

9-9866 

947 

8968  09 

849  278  123 

30.7734 

9.8201 

997 

994009 

991  026  973 

31-5753 

9-9900 

948 

89  87  04 

851  97i  392 

30.7896 

9.8236 

998 

996004 

994  oil  992 

3:-  S9" 

949 

900601 

854  670  349 

30.8058 

9.8270 

993 

99  80  01 

997  002  999 

31  .  6070 

9.9967 

950 

90  25  oo 

857  375  ooo 

30.8221 

9-8305 

IOOO 

I  OO  OO  OO 

I  OOO  OOO  OOO 

31.6228 

10.  OOOO 

72  PRACTICAL  SURVEYING 

MEASURES  OF  LENGTH  AND  AREA. 

12  inches I  foot  =  0.3047973  meter. 

3  feet I  yard  =  36  ins.  =  0.9143919  meter. 

5^  yards.  ...'.....  I  rod,  pole  or  perch  =  i6j  ft.  =  198  ins. 

40  rods I  furlong  =  J  mile  =  220  yds.  =  660  ft. 

8  furlongs I  statute  mile  =  320  rods  =  1760  yds. 

=  5280  ft. 

3  miles I     league  =  24     furlongs  =  960     rods 

=  5280  yds.  =  15,840  ft. 

Gunter's  chain  for  surveyors  =  66  ft.  =  4  rods  =  100 
links. 

I  link  =  0.66  ft.  =  7.92  ins. 
I  section  of  land  =  I  sq.  mile  =  640  acres. 
I   acre  contains  43,560  sq.   ft.  and  measures  208.71  X 
208.71  ft.  =  10  sq.  Gunter's  chains. 

The  vara  is  an  old  Spanish  measure  of  length.  It  is 
used  in  Mexico  and  in  some  of  the  western  states.  The 
legal  vara  in  California  =  33.372  ins.  In  San  Francisco, 
Cal.,  the  vara  =  33  ins.  The  vara  of  Castile  =  32.8748  ins. 

The  metric  system  is  decimal  and  is  based  on  the  meter 
(39.370428  ins.).  The  decimeter  =  TV  m.,  centimeter  = 
T£tf  m.,  millimeter  =  y^V^  m.  The  dekameter  =  10  m., 
hectometer  =  100  m.,  kilometer  =  1000  m.,  myriameter  = 
10,000  m.  The  metric  square  measures  have  the  word 
"square"  prefixed  to  the  measures  of  length  except  the 
square  hectometer  which  is  known  as  the  hectare  =  2.4711 
acres. 


CHAPTER    III 
LEVELING 

The  object  of  leveling  is  to  determine  the  difference  in 
elevation  between  two  or  more  points.  To  do  this  work 
requires  no  mathematical  knowledge  beyond  the  ability  to 
add  and  subtract. 

A  vertical  line  points  to  the  center  of  the  earth  and  a 
line  perpendicular  to  a  vertical  line  is  a  horizontal  line. 

A  level  line  is  parallel  with  the  surface  of  still  water 
and  each  point  marks  an  equal  distance  from  the  center 
of  the  earth.  In  plane  surveying  the  distances  between 
points  are  so  short  that  for  all  practical  purposes  a  hori- 
zontal line  is  considered  to  be  a  level  line. 

A  level  line  in  plane  surveying  is  one  having  the  same 
elevation  throughout  the  length  and  a  horizontal  line  is 
actually  only  a  line  of  apparent  level. 

In  Fig.  80  let  BE  represent  an  arc  of  the  A 
earth's  surface.  AD  represents  an  arc  about  B 
the  height  of  the  eye  of  an  observer  par-  ^^Ss.  '"IA 

allel   to  BE.     When  C  is  seen  from  A  the         pIG>  o0 
points  A  and  D  are  on  the  same  true  level 
and  the  points  A  and  C  are  on  the  same  apparent  level. 
In  a  distance  of  one  mile  the  difference  CD  is  practically 
8  ins. 

Let  D  =  distance  in  miles, 

h  —  difference  in  feet  between  true  and  apparent 
level, 

.       2D* 

then      h  = 

3 
When  D  =  distance  in  feet,  h  =  0.000,000,024  D2. 

Refraction  causes  objects  near  the  horizon  to  appear 
higher  than  they  are  actually.  For  very  long  sights, 

73 


74  PRACTICAL  SURVEYING 

especially  when  taken  early  or  late  in  the  day,  a  correction 
must  be  made  for  refraction.     The  formulas  then  become, 

when  D  =  distance  in  miles, 


9 

and,  when  D  =  distance  in  feetr 

h  =  0.000,000,021  £>2. 

With  instruments  ordinarily  used  and  with  usual  lengths 
of  sights  curvature  of  the  earth  and  refraction  cannot 
affect  the  work.  If  a  backsight,  however,  is  very  short 
and  a  foresight  is  very  long  both  the  aforementioned 
factors  may  have  a  slight  effect,  and  if  the  instrument  is 
not  in  good  adjustment  errors  will  be  multiplied  in  the 
proportion  the  length  of  backsight  bears  to  the  length  of 
foresight.  The  instrument  should  be  set  as  nearly  as 
possible  equidistant  between  points  on  which  the  rod  is 
held.  This  is  important. 

Very  simple  leveling  instruments  were  used  by  surveyors 
and  engineers  in  early  times,  and  are  just  as  useful  today 
when  well-made  modern  instruments  are  not  available. 

The  miner's  triangle.  —  The  early 
miners  in  California  set  grade  pegs  on 
hundreds  of  miles  of  ditches  and  roads 
with  this  primitive  instrument.  When 
the  grade  pegs  were  set  at  intervals 
of  one  rod  (i6J  ft.)  the  distance  from 
D  to  E  =  i  rod.  When  this  made 
the  triangle  inconveniently  large  the 
FiG.Si.  Miner's  triangle.  distance  was  8J  ft>>  or  IO  ft 

To  adjust  the  triangle  two  pegs  were  driven  so  the  ends 
of  the  triangle  could  rest  on  them,  the  tops  of  the  pegs 
being,  as  nearly  as  the  eye  could  judge,  at  the  same  ele- 
vation. From  the  apex  of  the  triangle  a  plumb-bob  was 
suspended  by  a  thread  or  fine  cord  and  the  feet  of  the  tri- 
angle placed  on  the  pegs. 

The  point  where  the  plumb  line  crossed  the  brace  was 
then  marked  (say  at  A).  The  ends  of  the  triangle  were 
then  reversed,  and  the  place  where  the  line  crossed  on  this 
trial  was  marked  (B).  Halfway  between  A  and  B  a 


LEVELING  75 

mark  C  was  cut  on  the  brace.  Whenever  the  triangle  was 
held  so  the  vertical  plumb  line  crossed  the  brace  at  C,  the 
ends  D  and  E  were  at  the  same  elevation. 

To  run  a  grade  line  with  a  miner's  triangle.  —  Suppose 
the  grade  is  to  be  |  in.  in  one  rod.  At  E  drive  a  nail 
with  the  head  projecting  \  in.  When  the  first  grade  peg 
is  driven  rest  the  leg  D  on  it  and  swing  the  triangle  until 
E  rests  on  the  ground  and  the  plumb  line  crosses  the  mark 
at  C.  Swing  E  far  enough  to  one  side  to  permit  a  peg  to 
be  driven  where  the  end  E  had  touched  the  ground.  Now 
place  E  so  the  nail  head  rests  on  the  peg  and  drive  the  peg 
until  the  plumb  line  crosses  C.  The  top  of  the  peg  under 
E  will  be  J  in.  lower  than  the  top  of  the  peg  under  D. 

To  run  a  grade  with  a  carpen- 
ter s  level  and  straight-edge.  — 
A  straight-edge  is  more  con- 
venient  to  use  than  a  miner's 
triangle,  and  is  used  in  the  FlG.  82>  straight-edge  and  level, 
same  manner  on  smooth  land 

and  in  ditches.  When  the  ground  is  covered  with  rocks 
or  vegetation  the  triangle  is  better  although  clumsy.  A 
straight-edge  is  usually  made  from  a  2-in.  plank.  In 
the  middle  for  a  length  of  2  or  3  ft.  the  depth  is  about 
7  ins.  and  at  the  ends  about  3  ins.  The  bottom  is  first 
made  perfectly  straight  after  which  the  top  is  made  parallel 
with  it.  A  carpenter's  level  is  used  on  top,  this  being- 
more  convenient  than  a  plumb-bob  and  line.  The  grade 
nail  is  driven  in  the  bottom  at  one  end  when  the  edges 
have  been  made  truly  parallel. 

To  adjust  a  straight-edge  so  the  top  and  bottom  will  be 
parallel.  —  First  test  the  level  and  be  certain  the  bubble  is 
in  correct  adjustment.  There  being  no  grade  nail  in  the 
bottom  of  the  straight-edge,  drive  a  peg  at  each  end  and 
with  the  carpenter's  level  held  on  top  drive  the  pegs  until 
the  bubble  indicates  the  tops  of  the  pegs  to  be  at  the  same 
elevation. 

Mark  the  glass  at  the  end  of  the  bubble  and  holding  the 
level  in  position  reverse  the  ends  of  the  straight-edge, 
resting  them  on  the  pegs.  The  bubble  will  move  and  the 
new  position  of  the  end  is  to  be  marked.  Midway  be- 
tween the  two  marks  make  a  third  mark.  Holding  the 


76  PRACTICAL  SURVEYING 

level  in  place  and  the  straight-edge  on  the  pegs  drive  down 
the  high  peg  gently  until  the  end  of  the  bubble  moves  to 
the  middle  mark. 

The  tops  of  the  pegs  are  now  at  exactly  the  same  ele- 
vation. Holding  the  straight-edge  in  place  reverse  the 
level  and  plane  down  the  high  end  of  the  top  of  the  straight- 
edge until  the  end  of  the  bubble  touches  the  middle  mark. 
The  bubble  in  the  carpenter's  level  should  remain  in  the 
middle  no  matter  how  the  level  is  placed  on  top  of  the 
straight-edge. 

Road  percenter.  —  This  instrument  was  in  common  use 
for  several  centuries  and  the  author  once  used  one  in  lay- 

B    m&  out  a  roa<^  ^or  a  mmmg  com- 

1     pany  when   no  better   instrument 

1     was  available.    The  same  principle 

is  seen  today  in  the  straight-edges 
used  by  brick  masons  for  plumbing 
walls. 

A  piece  of  wood  2  ins.  by  4  ins.  is 
made  perfectly  straight  on  top,  with 
pieces  of  tin  at  A  and  B  for  sights. 

Through  each  sight  at  points  the 
FIG.  83.     Road  percenter.      same  h?ight  aboye  the  tQp  of  the 

wood  a  hole  ^  in.  in  diameter  is  made.  At  D  a  i-in.  hole 
is  bored  through  for  the  upper  part  of  a  Jacob  staff.  A 
f-in.  by  3-in.  piece  15  ins.  long  is  attached  to  the  side  so 
that  a  line  scratched  down  the  middle  is  exactly  perpen- 
dicular to  the  top.  The  space  C  is  cut  out  and  a  12-oz. 
plumb-bob  hung  by  a  silk  thread  one  foot  long  attached 
near  the  top  swings  in  this  space. 

Theoretically  when  the  instrument  is  placed  on  the 
Jacob  staff  and  the  silk  plumb  line  covers  the  vertical 
scratch  the  line  of  sight  through  the  sighting  holes  is  hori- 
zontal. A  level  rod  is  used  with  this  instrument.  It  is 
termed  a  percenter  because  a  graduated  arc  is  sometimes 
placed  above  the  opening  in  which  the  plumb-bob  swings. 
When  the  eye  end  is  lowered  by  tipping  the  Jacob  staff 
until  the  plumb  line  crosses  the  2  per  cent  mark  the  line 
of  sight  instead  of  being  horizontal  is  inclined  2  per  cent 
to  the  horizontal.  Road  grades  are  expressed  in  rise  or 
fall  per  100  ft.,  that  is  in  per  cent  of  rise  per  foot  and  the 


LEVELING 


77 


angle  corresponding  to  any  per  cent  of  rise  being  known 
the  line  of  sight  may  be  set  at  the  proper  angle  and  the 
grade  stakes  set.  The  percenter  in  theory  is  good  but 
practically  is  of  little  service  except  on  straight  lines.  Used 
as  a  level  fairly  good  results  may  be  obtained  with  careful 
work. 

Water-tube  level.  —  This  level  consists  of  a  metal  tube 
two  or  three  feet  long  bent  up  at 

the  ends  with  glass  vials  proj ecting    (\ fss=_i j) 

above  the  ends  and  open  at  the  top. 
In  the  middle  a  socket  for  the  head 
of  a  Jacob  staff  or  a  tripod  is  fas- 
tened. The  tube  is  rilled  with  col- 
ored liquid  and  the  line  of  sight 
across  the  top  of  the  liquid  is  level. 


Water  tube  level. 


Leveling  with  rubber  hose.  —  For  leveling  line  shafts  in 
mills  where  it  is  often  difficult  to  use  levels  and  rods  the 
use  of  small  rubber  tubes  is  common.  One  end  of  the  tube 
is  fastened  to  a  tank  containing  water  with  the  surface  at 
nearly  the  elevation  of  the  shafting.  In  the  other  end  is 
a  graduated  glass  tube  corked  to  prevent  the  loss  of  water 
while  the  tube  is  being  carried.  Holding  the  glass  tube 
against  a  post  or  beam  on  which  a  mark  is  to  be  placed 
and  raising  it  until  the  top  of  the  water  is  a  few  inches  be- 
low the  end,  the  cork  is  removed.  The  surface  of  the  water 
when  the  cork  is  removed,  so  the  compression  of  enclosed 
air  will  have  no  effect,  will  be  level  with  the  surface  of  the 

water  in  the  tank  and  the 
height  can  be  marked  by  a 
cut  or  by  driving  a  nail.  A 
number  of  points  are  placed , 
from  which  the  millwright 
can  measure  to  level  the 
shafting. 

A  modern  level  consists 
of  a  telescope  mounted  in 
a  rigid  frame  together  with 
FIG.  85.     Dumpy  level— ordinary  type.  a  jQng  jevel  tube  containing 

a  sensitive  bubble.  By  means  of  right-and-left-threaded 
screws  the  bubble  is  brought  to  the  middle  of  the  tube. 
When  the  instrument  is  in  adjustment  the  bubble  remains 


PRACTICAL  SURVEYING 


stationary  while  the  frame  is  revolved  on  the  vertical 
axis  so  the  observer  may  look  in  any  direction  through  the 
telescope. 

A  ring  inside  the  telescope  tube  carries  cross-hairs,  or 
fine  wires,  to  define  the  line  of  sight.  The  wires  are  fo- 
cussed,  or  brought  into  the  field  of  view,  by  moving  the 


FIG.  86.     Dumpy  level  —  Bausch  and  Lomb  type. 

eyepiece.  The  line  of  sight  passes  through  the  center  of 
the  eyepiece  and  the  intersection  of  the  wires.  When  the 
instrument  is  in  adjustment  the  line  of  sight  is  horizontal 
when  the  level  bubble  is  in  the  middle  of  the  glass  tube, 
which  is  graduated  so  the  position  of  the  bubble  can  be 
located. 

In  Fig.  87  the  usual  arrangement  of  wires  is  shown  at 
(a).  The  vertical  wire  is  of  some  assistance  in  enabling 
the  leveler  to  tell  when  the  rod  is  vertical  one  way  while 

waving  the  rod  insures  vertical- 
ity  the  other  way.  To  assist  in 
obtaining  equal  backsights  and 
foresights  two  additional  wires  are 

A  B  c         sometimes  used  as  shown  at  (6) 

pIG   g7  and  (c).      The  horizontal  line  is 

caught  with  the  middle  wire  (b] 

and  the  upper  and  lower  wires  are  spaced  so  they  win 
intercept  one  foot  on  the  rod  at  a  distance  of  100  ft.,  two 
feet  at  a  distance  of  200  ft.,  etc.  Three  horizontal  wires 
are  used  by  some  men  to  prevent  mistakes  in  reading  the 
rod,  by  taking  readings  on  the  three  wires  at  each  turning 
point  or  bench  mark.  When  a  level  has  three  horizontal 


LEVELING  79 

wires  and  one  only  is  read  mistakes  often  occur,  so  the 
style  shown  at  (c)  is  sometimes  used.  With  this  arrange- 
ment the  rodman  holds  his  rod  horizontally  for  the  distance 
to  be  read,  after  which  he  sets  the  turning  point  and  holds 
the  rod  vertically  on  it. 

Three  kinds  of  levels  are  in  common  use  for  the  best  work 
and  they  are  known  as  the  wye  (Y),  dumpy  and  precision 
level.  The  latter  is  used  only  for  the  highest  grade  of 
government  work  and  is  made  as  a  form  of  wye  level  by 
some  makers  and  as  a  form  of  dumpy  level  by  others.  The 
latest  style  of  precision  level  is  of  the  dumpy  type  and  is 
known  as  the  United  States  Coast  and  Geodetic  Survey  level. 

The  wye  level  for  some  unexplained  reason  obtained  a 
strong  footing  in  the  United  States  nearly  a  century  ago. 
It  is  comparatively  easy  to  adjust,  but  the  adjustments  are 
those  of  the  shop  rather  than  the  field,  so  the  instrument 
requires  frequent  adjustment.  European  engineers  gen- 
erally express  surprise  when  they  first  learn  that  the  wye 
is  so  extensively  used  in  this  country.  The  author  sold 
his  wye  nearly  twenty  years  ago  and  purchased  a  dumpy, 
since  which  time  he  has  been  a  strong  advocate  of  the 
latter  type.  A  description  of  the  wye  level  and  its  adjust- 
ments is  given  in  every  instrument  maker's  catalogue.  A 
wye  level  may  be  adjusted  in  the  same  manner  as  a  dumpy. 

The  dumpy  was  so  named  because  the  earlier  ones  had 
inverting  telescopes  (that  is  all  objects  were  seen  upside 
down)  and  the  omission  of  one  set  of  lenses  called  for  short 
(dumpy)  tubes.  Inverting  telescopes  gather  more  light 
than  erecting  telescopes  and  are  used  for  government  work 
requiring  the  highest  precision.  For  most  of  the  work 
done  by  engineers  and  surveyors  nothing  is  gained  by 
using  an  inverting  telescope,  and  the  amount  of  training 
necessary  to  become  accustomed  to  viewing  objects  in  an 
unnatural  position  is  trying. 

The  dumpy  weighs  less  than  a  wye  of  the  same  power; 
costs  less;  retains  adjustments  longer  and  stands  rough 
usage  better. 

The  bubble  should  be  sensitive.  The  sensitiveness  of  a 
bubble  is  a  test  of  workmanship.  No  maker  will  put  a 
sensitive  bubble  on  a  poorly  made  instrument  and  a  slug- 
gish bubble  is  a  warning  to  the  prospective  purchaser. 


8o  PRACTICAL  SURVEYING 


TO  ADJUST  A  DUMPY  LEVEL 

1st  adjustment.  The  level  must  be  perpendicular  to  the 
vertical  axis. 

Set  the  level  up  by  spreading  the  tripod  legs  so  the  tele- 
scope will  be  about  five  feet  above  the  ground.  Push  each 
leg  into  the  ground  firmly.  The  two  plates  should  first  be 
made  parallel  by  means  of  the  leveling  screws  and  when 
the  instrument  is  set  up  the  plates  should  be  as  nearly 
horizontal  as  possible. 

Turn  the  telescope  so  it  is  over  two  screws.  Then  turn 
the  screws  together  and  bring  the  bubble  to  the  middle  of 
the  tube.  The  thumbs  move  towards  or  from  each  other 
but  never  move  in  the  same  direction.  The  bubble  travels 
in  the  direction  in  which  the  right  thumb  moves.  The 
screws  should  move  easily  without  binding  but  never 
loosely,  for  a  firm  seating  of  the  ends  is  necessary.  If 
screwed  tight  the  instrument  becomes  strained  and  the 
slightest  touch  disturbs  the  bubble.  Making  the  screws 
tight  also  injures  the  threads.  After  the  bubble  is  brought 
to  the  middle  over  one  pair  of  screws  revolve  the  telescope 
ninety  degrees  and  repeat  the  leveling  process  over  the 
other  pair  of  screws. 

The  foregoing  instructions  are  general  and  apply  to  the  use 
of  the  level  as  well  as  when  adjustments  are  being  tested. 

The  level  having  been  brought  to  the  middle  of  the  tube 
over  each  pair  of  screws  turn  the  telescope  end  for  end 
over  one  pair  and  if  the  bubble  remains  in  the  middle  the 
level  is  perpendicular  to  the  vertical  axis.  If  it  moves 
away  from  the  middle  bring  it  halfway  back  by  means  of 
one  pair  of  screws  and  the  rest  of  the  way  by  turning  the 
capstan  head  screws,  on  one  end  of  the  level  tube,  with  an 
adjusting  pin.  Then  level  it  over  the  other  pair  and  test. 

2nd  adjustment.  The  horizontal  wire  must  be  perpendic- 
ular to  the  vertical  axis. 

The  instrument  maker  fixes  the  wires  in  the  reticule  per- 
pendicular to  each  other.  Hang  a  heavy  plumb-bob  in  a 
sheltered  place  so  the  plumb  line  will  not  move.  After 
adjusting  the  bubble  sight  on  the  plumb  line.  If  the 
vertical  wire  covers  it  the  horizontal  wire  is  perpendicular 
to  the  vertical  axis. 


LEVELING 


8l 


If  the  wire  and  plumb  line  form  a  small  angle  loosen 
the  capstan  head  screws  on  the  sides  of  the  telescope,  hold- 
ing the  reticule,  and  tap  lightly  with  the  finger  until  the 
reticule  is  shifted  enough  to  bring  the  wire  into  a  vertical 
position.  Then  tighten  the  screws.  Fig.  88. 

$rd  adjustment.  The  line  of  sight  must  be  perpendicular 
to  the  vertical  axis. 

After  the  instrument  has  been  tried  for  the  first  and 
second  adjustments,  and  the  adjustments  made,  drive  two 


FIG.  89. 


FIG.  90. 


stakes  from  200  to  300  ft.  apart  and  set  the  level  exactly 
midway. 

Level  it  carefully  and  read  the  rod  held  on  A.  The  rod 
is  next  held  on  B  and  a  reading  taken  with  the  instrument 
level,  that  is  with  the  bubble  in  the  middle  of  the  tube. 
The  difference  of  the  readings  is  the  difference  in  elevation 
between  A  and  B,  no  matter  how  badly  the  instrument  may 
be  out  of  adjustment. 

The  instrument  is  then  set  as  close  as  possible  to  a  rod 
held  vertically  on  one  stake  with  the  eyepiece  next  to  the 
rod.  The  bubble  is  brought  to  the  middle  of  the  tube  and 
the  observer  looking  through  the  object  glass  focuses  the 
telescope  until  he  can  read  the  rod,  which  appears  to  be  at 
a  great  distance.  The  cross-hairs  cannot  be  seen  so  the 
elevation  is  taken  in  the  center  of  the  field  of  view,  a  lead 
pencil  or  pointed  stick  marking  the  point. 

Assuming,  for  example,  that  the  instrument  is  placed  in 
front  of  stake  A ,  and  that  B  is  lower  than  A ,  add  the  differ- 


82  PRACTICAL  SURVEYING 

ence  to  the  rod  reading.  If  B  is  higher  than  A  subtract  the 
difference  in  elevation  from  the  rod  reading.  Set  the  target 
at  the  height  thus  obtained  and  have  the  rodman  hold  the 
rod  on  B. 

The  effect  of  the  curvature  of  the  earth  and  refraction, 
assuming  the  distance  of  B  from  A  to  be  250  ft.,  will 
amount  to  0.000,000,021  X  25O2  =  o.ooi  ft.,  by  which 
amount  the  target  should  be  lowered  if  the  correction  is 
applied.  Errors  in  observation  will  probably  offset  this 
small  correction  which  may  therefore  be  disregarded. 
*  The  rod  being  held  vertically  on  B  the  telescope  is 
pointed  towards  the  target  and  the  leveling  screws  turned 
until  the  bubble  is  exactly  in  the  middle  of  the  tube.  By 
means  of  the  capstan  screws  on  the  top  and  bottom  of  the 
telescope  the  reticule  is  raised  or  lowered  until  the  hori- 
zontal wire  intercepts  the  center  of  the  target.  This  com- 
pletes the  adjustment.  This  adjustment  on  a  wye  level 
makes  the  wye  adjustment  unnecessary. 

THE  DATUM 

It  has  been  shown  that  a  level  is  merely  an  instrument 
by  means  of  which  a  horizontal  line  may  be  determined. 

The  horizontal  line  passing  through  the  center  of  the  eye- 
piece and  the  intersection  of  the  cross  wires  is  a  base,  but 
for  convenience  in  platting  and  computing,  this  base  is 
assumed  to  be  at  some  definite  height  above  a  parallel 
base  termed  a  "datum,"  or  "datum  plane."  An  "arbi- 
trary datum"  is  one  arbitrarily  chosen  for  a  particular 
piece  of  work  and  is  often  assumed  as  being  100  ft.  below 
the  line  of  sight  at  the  starting  point. 

In  city  work  the  datum  is  usually  assumed  to  be  100  ft. 
below  the  lowest  point  on  the  streets  of  the  city.  When- 
ever any  Government  bench  marks  are  close  by  the  sur- 
veyor uses  the  elevation  marked  on  the  nearest  one,  the 
Government  datum  (o)  being  the  mean  of  low  tides.  In 
some  seacoast  cities  the  mean^of  lower-low  tides  is  used  as 
datum.  If  it  is  believed  that  excavations  will  be  made  below 
the  Government  datum  it  is  best  to  use  an  elevation  of 
100  ft.  for  the  datum  instead  of  o  to  avoid  mistakes  likely 
to  arise  when  minus  elevations  are  used.  This  in  effect 


LEVELING  83 

fixes  an  arbitrary  datum  one  hundred  feet  below  the  mean 
of  low  tides. 

If  a  hole  is  dug  ten  feet  below  Government  datum  then 
the  elevation,  referred  to  datum,  is  — 10.  A  wall  lo-ft. 
high  with  the  bottom  at  datum  will  have  an  elevation  on 
top  of  +10.  The  use  of  the  -f-  and  —  signs  has  been  a 
fruitful  source  of  error  and  the  simple  expedient  of  assum- 
ing the  zero  datum  below  the  lowest  point  an  excavation 
may  reach  stops  all  trouble. 

In  the  example  just  cited  assume  that  an  arbitrary  datum 
has  been  selected  100  ft.  below  the  Government  datum. 
The  bottom  of  the  hole  will  have  an  elevation  of  looft. 
above  datum  and  the  top  of  the  wall  will  have  an  elevation 
of  no  ft.  above  datum. 

LEVEL  RODS 

There  are  several  types  of  level  rods  known  as  Boston, 
New  York,  Philadelphia,  etc.     A  description 
of  each  rod  is  not  necessary  for  this  informa- 
tion is  contained  in  the  catalogues  of  instru- 
ment makers. 

Rods  graduated  with  thin  black  lines  on 
varnished  wood  on  which  a  target  is  neces- 
sary are  not  much  used  today.  The  favorite 
type  is  some  form  of  Philadelphia  rod.  This 
is  known  as  "self-reading"  because  a  target 
is  not  necessary.  The  face  of  the  rod  is 
painted  white  and  originally  was  graduated 
in  feet  and  tenths  of  a  foot.  When  closer 
readings  were  wanted  a  scale  on  the  sliding 
target  was  used  by  means  of  which  half-hun- 
dred ths  could  be  read. 

The  rod  was  later  made  with  graduations 
of  one-hundredth  of  a  foot  stenciled  on  the 
face,  the  marks  being  alternately  black  and 
white.  A  target  with  scale  was  attached  to 
the  rod  so  that  readings  could  be  taken  to  FIG.  91.  Phila- 
the  nearest  half-hundredth.  An  improve-  delphia  self- 
ment  was  made  when  the  graduations  were  reading  rod. 
cut  into  the  white  face  of  the  rod  with  the  alternate  gradua- 
tions painted  black.  Another  form  has  the  graduations  cut 


84  PRACTICAL  SURVEYING 

without  the  alternate  black  and  white  spaces.  The  author, 
and  probably  the  majority  of  engineers  and  surveyors, 
prefers  the  broad  marks,  as  the  reading  of  the  rod  does  not 
tax  the  eyes  and  it  is  "self-reading"  at  long  distances.  A 
target  with  a  vernier  reading  to  a  thousandth  of  a  foot  is 
used  on  all  rods  except  those  with  a  stenciled  face. 

The  vernier  is  a  scale  attached  to  the  target.  It  is 
divided  into  ten  equal  divisions  with  the  zero  on  the  center 
line  of  the  target.  The  ten  divisions  on  the  vernier  cover 
nine  divisions  (nine  one-hundred ths)  on  the  rod,  each 
division  on  the  vernier  being  thus  one-tenth  of  one-hun- 
dredth, or  one- thousandth,  of  a  foot  less  than  the  smallest 
graduation  on  the  rod. 

To  read  less  than  one-hundredth  of  a  foot,  read  upward  on 
the  rod  until  the  hundredth  below  the  zero 
of  the  vernier  is  reached.  Then  read  up- 
wards from  zero  on  the  vernier  until  a  line 
is  reached  that  coincides  with  some  line  on 
the  rod.  The  vernier  in  Fig.  92  reads  6  feet, 
I  tenth,  2  hundredths  and  4  thousandths 
(6.124)  and  is  read  "six  point  one,  two, 
four." 
FIG.  92.  Vernier  A  level  rod  is  merely  a  rod  graduated  in 

tarret^1  ^  feet'  tenths  and  hundred  ths  of  a  foot;  or  in 
feet,  inches  and  eighths  of  an  inch,  the  divi- 
sion in  inches  being  used  only  by  architects  and  building 
mechanics.  The  decimally  divided  foot  is  used  by  en- 
gineers and  surveyors. 

The  graduations  proceed  upward,  the  foot  of  the  rod  being 
zero.  It  is  customary  to  mark  the  feet  in  red,  the  figures 
being  "one- tenth"  high  and  the  tenths  in  black,  the  figures 
being  a  half-tenth  high.  Each  five-hundredth  mark  pro- 
jects slightly  beyond  the  marks  denoting  hundredths. 

When  a  level  is  set  up  it  may  be  turned  in  any  direction 
and  a  rod  is  used  to  determine  the  height  of  the  horizontal 
line  above  the  ground.  The  rod  must  be  held  perfectly 
vertical,  or  plumb,  and  wherever  the  horizontal  line  of 
sight  strikes  it  a  "reading"  is  obtained. 

A  rodman  must  always  stand  directly  back  of  the  rod 
and  face  the  leveler,  so  the  latter  will  see  the  face  of  the 
rod  fully  and  not  at  an  angle.  The  bottom  of  the  rod  must 


LEVELING 


rest  firmly  on  the  point  on  which  it  is  held.  The  hands 
should  be  about  the  height  of  the  chin  to  support  the  rod 
properly  and  the  fingers  should  grasp  the  sides,  never  en- 
circling the  rod,  for  they  will  cover  some  of  the  graduations 
and  may  interfere  with  a  sight  if  placed  across  the  face. 


FIG.  93.    Rod  level. 


FIG.  94.    Folding  rod  level. 


For  holding  the  rod  in  a  vertical  position  several  forms 
of  rod  levels  are  on  the  market.  Some  men  use  a  light 
plumb-bob,  generally  a  nuisance  in  a  wind.  A  time-honored 
method  is  " waving  the  rod." 

In  Fig.  95  A-B  is  the  horizontal  line  and  C-D  the  rod. 
When  the  rod  is  held  at  C  and  allowed  to  lean  towards  D' 
or  fall  back  towards  D" ',  the  horizon- 
tal line  gives  a  greater  reading  than  A 
when  C— D  is  perpendicular  to  A—B. 
When  the  rodman  has  no  level  or 
other  means  for  insuring  a  vertical 
position  of  the  rod  and  holds  the 
rod  on  a  point  on  which*  a  close  read- 
ing is  desired,  the  rod  is  waved  to- 
wards and  from  the  leveler  until  he 
obtains  the  shortest  reading  possible. 


FIG.  95. 


The  "waving"  must  be  done  very  slowly.  Some  instru- 
ment makers  furnish  targets  made  in  the  shape  of  an  angle 
and  the  horizontal  division  line  on  the  target  cannot  be 
completely  covered  by  the  cross  wire  unless  the  rod  is  ver- 
tical. This  necessitates  signaling  by  the  leveler  and  the 
author  believes  waving  a  rod  on  particular  points  cannot 
be  improved  upon.  If  a  rodman  is  not  experienced  he 
should  have  a  rod  level  for  intermediate  sights. 

A  rod  is  read  with  target  to  the  nearest  "thousandth" 
on   bench    marks;     to   the   nearest    "half-hundredth"   on 


86 


PRACTICAL  SURVEYING 


turning  points,  and  the  nearest  "tenth"  for  intermediate 
points.  The  use  of  the  target  on  turning  points  is  optional 
but  is  customary  so  the  reading  by  the  leveler  can  be 
.checked  by  the  rodman.  The  target  is  never  used  on 
intermediate  sights  or  for  "stations"  except  when  grade 
pegs  are  set.  In  setting  grades  it  is  very  convenient  and 
easy  on  the  eyes  to  use  a  target. 


FIG.  96.     Angle  target. 

Checking  a  reading  is  done  as  follows :  The  leveler  reads 
the  rod  and  records  the  reading.  He  then  motions  to  the 
rodman  to  set  the  target  and  directs  him  in  moving  it  up 
or  down  until  it  is  practically  right.  Then  the  rod  is 
waved  and  the  target  moved  until  it  is  set,  when  the  rod- 
man clamps  it.  He  then  reads  the  rod  and  compares  his 
reading  with  that  of  the  leveler  as  the  latter  goes  past  to 
a  new  set  up.  When  the  backsight  is  taken  he  gives  his 
reading  to  the  leveler  as  a  check  before  going  ahead  on 
line. 

When  the  leveler  wishes  to  have  the  target  used  he  raises 
his  right  arm  vertically  as  high  as  possible  and  describes 
a  small  horizontal  circle.  This  means  "turning  point." 
The  rodman  selects  a  good  point,  holds  the  rod  on  it,  slides 
the  rod  to  where  he  thinks  the  line  of  sight  will  intersect 
it  and  awaits  orders.  The  target  is  moved  up  when  the 
leveler  puts  his  right  hand  out  horizontally  to  the  side. 
It  is  moved  down  when  he  holds  out  his  left  hand.  He 
thrusts  out  both  arms  when  it  is  right.  When  the  rod  is 
to  be  waved  he  holds  up  his  right  arm  and  waves  it  forward 
and  back. 

For  readings  on  line  at  stations  the  rod  is  held  on  the 
ground.  Turning  points  must  be  hard  and  so  firm  they 


LEVELING  87 

will  not  move  when  the  rod  is  held  on  top.  In  stony  coun- 
try the  tops  of  stones  make  excellent  turning  points  and  in 
timbered  land  exposed  roots  and  notches  in  trees  are  used. 
In  some  sections  where  the  soil  is  light  or  sandy  the  rod- 
man  carries  pegs  to  drive  in  the  ground  for  use 
as  turning  points. 

A  convenient  turning  point  consists  of  an 
iron  pin  attached  by  a  light  chain  to  the 
rodman's  belt.  The  pin  is  pushed  into  the 
ground  and  the  rod  held  on  top  of  the  ring. 
The  chain  is  used  to  keep  the  rodman  from 
going  off  without  the  pin  after  holding  the  rod 
on  it. 

To  find  the  difference  in  elevation  between  two 
points  not  far  apart. 

Assume  the  points  to  be  so  close  together       FtG  97 
that  they  may  both  be  seen  from  an  inter- 
mediate point.     Let  A  and  B  be  the  points  on  the  surface  of 
the  earth  and  C  and  D  be  points  vertically  above  them  on  a 
true  level  with  F,  on  the  line  of  sight. 

If  F  (position  of  level)  is  midway  between  C  and  D  the 
difference  between  true  level  and  apparent  level  will  be  the 
same.     That  is 


CA  ^EA 
BG 


FIG.  98.  If  the  level  is  not  midway  between 

A  and  B  it  will  be  necessary  to  meas- 

ure the  distances  AH  and  BH  and  use  the  formulas  al- 
ready given  to  determine  the  difference  between  the  true 
level  on  the  arc  CFD  and  the  apparent  level  on  the  hori- 
zontal line  EFG.  The  difference  in  elevation  is  obtained 
by  setting  the  instrument  up  at  some  point  H  between  A 
and  B  and  leveling  it  so  the  line  of  sight  EFG  is  hori- 
zontal. A  graduated  rod  is  held  vertically  at  A  and  the 
reading  taken  where  the  line  of  sight  intersects  the  face  of 
the  rod.  The  rod  is  then  held  vertically  at  B  and  a  read- 
ing taken.  The  difference  between  these  readings  gives  the 
difference  in  elevation  between  the  two  points,  provided 
they  differ  in  elevation* 


88 


PRACTICAL  SURVEYING 


PROBLEMS 

1.  Let  AH  =  240  ft.  and  BH  =  300  ft.     AE  =  10  ft., 
BG  =  6  ft.     What  is  the  true  difference  in  elevation  be- 
tween A  and  B?     If  AH  =  BH  what  will  be  the  true  differ- 
ence in  elevation? 

2.  Let  AH  =  1040  ft.  and  BH  =  1820  ft.     AE  =  5  ft., 
BG  =  6  ft.     Find  true  and  apparent  difference  in  elevations. 

3.  Let  AH  =  792  ft.  and  BH  =  ii88ft.     AE  =3.17  ft., 
BG  =  5.67    ft.      Find    true    and    apparent   difference  in 
elevations. 

DIFFERENTIAL   LEVELING 

To  find  the  difference  in  elevation  between  two  points  far  apart. 
The  line  ABCD  is  a  profile  of  the  ground.     The  differ- 
ence in  elevation  between  A  and  D  is  wanted  and  the  points 

are  assumed  to  be  about  2000 
ft.  apart.  Since  sights  should 
not  be  taken  more  than  400 
ft.  and  the  level  should  be  as 
nearly  as  possible  equidistant 
from  turning  points  the  level 
must  be  set  up  three  times. 
The  leveler  starts  from  A  and  walks  in  the  direction  of 
D,  counting  his  paces  until  he  is  about  100  paces  from  A. 
Going  to  one  side  of  the  line  until  he  believes  a  horizontal 
line  will  be  above  the  ground  at  A,  he  sets  up  the  level  and 
levels  the  telescope.  The  elevation  of  A  is  assumed  to  be 
some  definite  distance  above  datum  —  if  the  actual  eleva- 
tion is  not  known  —  and  the  rodman  holds  his  rod  on  A . 

The  leveler  reads  the  rod  and  puts 
the  rod  reading  (a)  in  his  field  book  in 
a  column  headed  by  a  -f-  sign.  The 
rodman  now  paces  from  A  to  E  and  an 
equal  number  of  paces  beyond  E  to  B 
where  he  selects  a  good  turning  point 
and  holds  his  rod.  The  leveler  reads 
the  rod  and  puts  the  rod  reading  (b)  in 
the  column  headed  by  a  —  sign.  He 


FIG.  99. 


Sta. 

+ 

- 

A 

3 

o 

B 

6 

9 

C 

4 

2 

D 

o 

3 

13 

14 

13 

I 

now  goes  to  Fand  levels  the  instrument,  - 

the  rodman  holding  the  rod  on  the  turning  point  (or  turn, 

as  it  is  commonly  called)  but  turning  it  so  the  leveler  can 


LEVELING  89 

see  the  face.  The  leveler  takes  the  reading  (bl)  and  records 
it  in  the  +  column  while  the  rodman  paces  to  Fand  an  equal 
distance  beyond  and  holds  the  rod  on  a  turn  at  C.  The 
leveler  now  reads  the  rod  and  puts  the  reading  (c)  in  the  — 
column,  then  sets  the  level  at  G,  judging  as  closely  as  pos- 
sible with  the  eye,  for  no  great  accuracy  is  required,  that  the 
distance  from  C  to  G  =  G  to  D.  He  then  reads  the  rod 
and  puts  the  reading  (cr)  in  the  -f  column.  The  rodman 
then  holds  the  rod  on  D  and  the  leveler  places  the  reading 
(d)  in  the  —  column. 

The  -f-  readings  and  —  readings  are  separately  added 
and  the  difference  between  them  is  the  difference  in  eleva- 
tion between  A  and  D.  The  forward  point  is  lower  than 
the  starting  point  if  the  +  readings  are  less  than  the  — 
readings.  It  is  of  course  higher  if  the  sum  of  the  —  read- 
ings exceeds  the  sum  of  the  -f-  readings. 

The  rod  readings  given  in  the  example  are  marked  in  the 
figure  for  identification.  The  profile  is  not  drawn  in  the 


FIG.  100.    Bench  mark  on  tree. 

field  book,  the  record  of  readings  on  the  ruled  page  being 
sufficient.  A  "backsight"  is  a  +  reading  and  a  "fore- 
sight" is  a  —  reading.  The  intermediate  points  are  not 
marked  in  differential  leveling,  the  leveler  entering  on  his 
book,  instead  of  B  and  C,  the  letters  "T.  P. "  (turning  point) 
or  "Peg,"  the  latter  word  being  a  survival  from  the  days 
when  every  turning  point  was  actually  a  peg  left  in  the 
ground. 

BENCH  MARKS  AND   TURNING  POINTS 

A  bench  mark  is  a  permanent  point  with  a  recorded  eleva- 
tion. In  the  country  a  stone  is  used  when  available  or  a 
tree  is  cut  and  marked.  A  cut  is  made  in  the  side  as  if  for 
a  broad  blaze.  The  lower  part,  however,  is  cut  in  like  a 


90  PRACTICAL  SURVEYING 

shelf  and  sloped  each  way  from  a  ridge  in  the  middle  so 
there  will  be  a  point  on  which  to  hold  the  rod.  At  the  edge 
a  nail  or  spike  is  driven  flush.  If  the  head  projects  it  may 
be  hammered  down  and  the  elevation  thus  disturbed. 
The  object  in  driving  the  spike  is  to  furnish  a  hard  point 
on  which  to  hold  the  rod.  The  bench  mark  may  be  used 
for  years  and  if  the  wood  rots  the  elevation  will  not  be 
preserved  unless  a  metal  point  is  used. 

On  some  smooth  place  as  close  as  possible  to  the  bench 
mark  the  elevation  should  be  marked.  When  trees  are 
blazed  the  figures  are  cut  into  the  wood  with  a  surveyor's 
scribe,  which  is  a  useful  tool  sold  by  all  instrument  dealers. 
An  accurate  description  of  each  bench  mark  is  written  in 
the  level  book  and  recorded  in  the  office. 

On  route  surveys  bench  marks  are  placed  about  one-third 
of  a  mile  apart  as  close  as  possible  to  the  line  but  far  enough 
to  the  side  to  be  preserved  during  the  construction  period. 
Turning  points  are  merely  temporary  benches  and  no  record 
is  usually  made  of  them.  For  the  most  accurate  work  the 
distance  between  turning  points  should  be  not  less  than 
200  ft.  (100  ft.  from  the  instrument)  nor  more  than  600 
ft.  For  ordinary  work  the  distance  between  turning 
points  should  not  exceed  1200  ft.  in  the  middle  of  a  com- 
fortable day.  Sights  should  be  short  in  the  early  morning 
and  late  afternoon;  also  on  very  bright,  hot  days.  The 
instrument  must  be  level,  in  perfect  adjustment  and 
midway  between  turning  points.  The  rod  must  be 
vertical. 

In  cities  or  on  construction  work  there  cannot  be  too 
many  bench  marks.  Corners  of  iron  or  masonry  steps; 
curb  stones;  tops  of  hydrants,  etc.,  are  used  and  recorded. 
Always  have  two  benches  close  together  on  such  work  and 
read  on  both.  The  records  give  the  difference  in  elevation 
of  the  benches  and  if  this  is  not  verified  then  one  must 
have  been  disturbed  and  each  must  be  compared  with 
some  near-by  bench  until  two  are  found  to  agree  with  the 
records.  These  are  used  and  the  faulty  ones  removed 
from  the  records. 

Errors  in  adjustment,  effect  of  the  curvature  of  the  earth 
and  effect  of  refraction  are  taken  out  by  equality  of  sights 
to  turning  points.  A  difference  of  one-tenth  the  length  of 


LEVELING  QI 

the  average  sight,  provided  the  differences  are  about 
equally  plus  and  minus,  will  not  affect  the  work  noticeably. 

The  rod  may  be  held  approximately  vertical  on  inter- 
mediate sights  for  ground  reading  but  lack  of  verticality 
on  bench  marks  and  turning  points  causes  errors  of  a  cumu- 
lative nature. 

The  sun  heats  the  end  of  the  level  toward  it  and  the 
metal  expanding  raises  that  end.  The  bubble  always 
seeks  the  high  end  and  this  error,  which  is  cumulative,  can 
only  be  guarded  against  by  bringing  the  bubble  to  the  center 
after  setting  the  target,  and  then  obtaining  a  new  reading. 
On  very  careful  work  the  instrument  should  be  shaded. 

Errors  in  reading,  of  one  foot,  one-tenth,  etc.,  are  often 
made,  but  are  more  common  with  target  rods  than  with 
self-reading  rods.  Requiring  the  rodman  to  carry  a  book 
for  "peg  readings"  so  he  can  check  the  leveler  is  a  practice 
that  should  never  be  omitted.  The  only  real  check  is  to 
duplicate  the  work  in  an  opposite  direction,  so  on  all  route 
work,  or  "profile  leveling,"  the  leveler  "checks  benches" 
every  ten  miles  and  sometimes  for  shorter  distances. 

All  persons  using  instruments  carry  their  personality 
into  their  work  so  a  recognized  difference  in  results  obtained 
by  different  individuals  has  been  termed  by  scientific  men 
the  "personal  equation."  With  increase  in  experience 
errors  due  to  personality  become  small  and  tend  to  balance, 
thus  eliminating  error  due  to  this  cause.  The  personal 
equation  shows  up  when  a  leveler  checks  on  a  bench  set 
by  another  man  before  the  personal  errors  of  either  have 
balanced. 

Careful  leveling  is  a  continual  balancing  of  small  errors 
but  the  residual  errors  are  cumulative.  To  duplicate  a 
line  of  levels  in  an  opposite  direction  does  not  change  the 
sign  (+  or  — )  of  the  residual  error.  The  error  increases 
with  distance  so  to  run  a  line  of  levels  ten  miles  and  re-run 
it  in  the  opposite  direction  is  the  same  as  running  one  line 
twenty  miles.  All  work  should  be  thus  checked  when 
possible  and  one-half  the  error  found  at  the  starting  point 
should  be  used  as  a  correction  to  the  bench  elevation  at  the 
other  end  and  proportionately  to  intermediate  benches. 

Failures  to  make  a  close  check  when  one  route  survey 
crosses  another  and  a  leveler  reads  a  bench  set  by  another 


PRACTICAL  SURVEYING 

man  often  happens  because  rods  of  different  makers  were 
used  without  comparing  the  graduations.  When  a  num- 
ber of  rods  are  used  they  should  be  purchased  under  good 
specifications  from  one  manufacturer  and  before  sending 
to  the  field  should  be  carefully  tested  for  accuracy  of  the 
graduations  with  a  standard  steel  tape. 

On  long  lines  of  levels  the  error  increases  as  the  square 
root  of  the  distance,  while  on  short  lines  the  error  is 
much  less.  In  city  work  benches  should  be  established  if 
possible  on  the  corner  of  each  block,  in  pairs,  and  in  the 
middle  of  long  blocks.  The  limit  of  error  in  feet  should 
never  exceed  the  following: 

e  —  a  Vm, 

in  which  e  =  error  in  feet, 
a  =  error  factor, 
m  =  distance  in  miles  between  the  benches. 

For  city  work  and  all  accurate  leveling  a  =  0.017. 

For  ordinary  route  leveling  a  =  0.034. 

When  leveling  down  hill  time  is  saved  if  the  horizontal 
line  is  brought  close  to  the  foot  of  the  rod  for  a  backsight. 
When  going  up  hill  the  horizontal  line  should  be  brought 
close  to  the  top  of  the  rod.  It  often  happens  that  an  at- 
tempt to  read  near  one  end. of  the  rod  results  in  the  hori- 
zontal line  striking  below  or  above  the  rod,  so  the  level 
must  again  be  set  up.  To  avoid  this  the  leveler  should 
have  a  hand  level  by  means  of  which  he  can  obtain  a  hori- 
zontal line  at  the  height  of  his  eyes.  The  level  is  then  set 
up  with  the  telescope  at  this  height  and  leveled. 

ROUTE,   OR  PROFILE,  LEVELING 

Differential  leveling  has  been  illustrated  and  such  work 
is  done  when  the  only  elevations  wanted  are  those  at  the 
ends  of  the  line,  every  intermediate  reading  being  on  a 
turning  point.  "Check  leveling,"  to  check  elevations  of 
bench  marks,  is  differential  leveling. 

Profile  leveling  is  for  the  purpose  of  making  a  profile 
and  obtain  a  record  of  heights  at  all  changes  of  elevation 
on  the  line. 

The  route  is  laid  off  in  stations  and  the  leveler  obtains 
the  elevation  of  the  ground  at  each  station,  and  also  at 


LEVELING 


93 


intermediate  points  where  decided  changes  occur,  as 
gullies,  etc.  Between  adjacent  T.  P.'s  there  may  be  sev- 
eral readings  and  on  steep  ground  there  may  be  several 
turns  in  one  station. 

The  level  is  set  up  and  a  reading  taken  on  a  convenient 
bench  mark.  This  reading  added  to  the  elevation  of  the 
bench  gives  the  elevation  of  the  horizontal  wire  and  is 
recorded  in  the  column  headed  H.  I.  (Height  of  Instrument). 
Readings  are  then  taken  on  the  ground  at  each  station  and 
-f-  station  and  subtracted  from  the  H.  I.,  the  difference  being 
the  ground  elevation.  A  reading  taken  to  a  bench  mark 
or  turning  point  to  get  the  H.  I.  is  known  as  a  backsight 
and  placed  in  the  B.  S.  column.  A  reading  taken  on  the 
ground  or  on  a  turning  point  or  bench  mark  to  obtain  the 
elevation  is  known  as  a  foresight  and  is  placed  in  the  F.  S. 
column.  The  notes  are  placed  on  the  left-hand  page  of  the 
field  book,  as  follows: 


Station. 

B.  S. 

H.I. 

P.  S. 

Elevation. 

Remarks. 

B.  M. 

o 

1.234 

ioi  .  234 

2    7 

100.00 

98  5 

B.  M.    on    oak   tree 
20  ins.  diam. 
ii  ft.  to  left  of  Sta.  i 

i 

e   4 

y     x 
qe   8 

+  50 

7.8 

07  .4 

4-85 

3  -° 

98.  2 

T.  P.  2  A 

2 

2.176 

102.723 

0.687 

I  .7C 

100.547 
IOI  .OO 

T.  P.  on  hub  at  Sta.  2 

P4eg 

2.  2 
8.431 

100.5 
94.292 

In  Fig.  ioi  is  illustrated  a  common  feature  in  profile 
leveling.  A  gully  runs  across  the  surveyed  line  and  the 
chainmen  when  setting  stakes  meas- 
ured the  -f-  distances  to  the  edges 
and  bottom.  When  the  levels  are 
taken  the  rodman  paces  the  distance 
from  the  station  to  the  bank  and  a 
reading  is  obtained. 

Leaving  his  level  the  leveler  by 

means  of  his  hand  level  obtains  the  elevation  at  the  bot- 
tom,  the  process   being  differential  leveling.      The  notes 


94 


PRACTICAL  SURVEYING 


are  placed  on  the  right-hand  page  of  the  level  book.  He 
returns  to  his  level  and  the  rodman  holds  on  the  opposite 
bank.  After  a  reading  is  taken  he 
paces  the  distance  to  the  next  station 
and  calls  it  to  the  leveler  who  then 
writes  down  the  +  stationing.  The 
three  +  distances  are  known  to  be 
only  closely  approximate  but  they 
check  the  distances  previously  meas- 
ured with  the  tape  and  so  identify  the  place.  In  rough 
country  the  hand  level  saves  much  time. 

The  notes  are  platted  on  profile  paper,  there  being  two 
rulings  in  common  use,  plate  A  with  20  and  plate  B  with  30 
horizontal  lines  to  an  inch.  Each  plate  has  four  vertical 


FIG.  102.    Two  types  of 
hand  levels. 


FIG.  103.    Plate  A  profile  paper. 

lines  to  an  inch.  When  plotting  the  profile  each  horizontal 
space  represents  a  station  and  each  vertical  space  repre- 
sents one  foot,  the  distorted  scale  showing  plainly  all  minor 
irregularities  in  the  surface.  Cuts  and  fills  and  differences 
in  elevation  are  easily  read  to  the  nearest  half  foot  and  es- 
timated to  the  nearest  quarter  of  a  foot.  For  estimating 
earthwork  quantities  this  is  convenient  but  the  principal 
value  of  the  exaggerated  vertical  scale  is  the  ease  with  which 
grade  lines  are  selected. 

The  purpose  in  making  a  profile  survey  is  usually  to 
select  the  grade  for  a  road,  railway  or  ditch.  When  the 
profile  is  made  a  thread  is  stretched  between  the  hands  and 
held  on  the  profile  so  the  projections  above  the  line  are 
equal  to  those  below.  In  common  language  the  "cuts  and 
fills"  balance.  If  this  gives  a  steeper  grade  than  the 


LEVELING  95 

maximum  allowed,  the  position  of  the  thread  must  be 
altered.  Two  considerations  therefore  enter  in  the  prob- 
lem of  fixing  a  grade  line,  one  being  economy  in  construction, 
the  other  economy  in  hauling  loads  up  the  grade.  In  irri- 


FIG.  104.     Plate  B  profile  paper. 

gation  ditches  the  grade  must  be  light  so  water  can  be 
delivered  to  the  highest  point  of  the  land  yet  steep  enough 
to  secure  a  velocity  that  will  prevent  the  deposit  of  silt  and 
the  growth  of  vegetation. 

In  drainage  ditches  the  grade  should  be  as  steep  as 
possible  without  producing  a  velocity  of  flow  great  enough 
to  scour  the  bottom  and  make  a  gully  of  the  ditch. 

TO   SET   GRADE   STAKES 

Contour  road  and  ditch  surveys  are  generally  made  with 
a  target  rod.  The  first  stake  is  driven  so  that  the  top  is 
on  grade.  The  fall  per  station  is  given  to  the  rodman. 
A  reading  is  taken  with  the  rod  held  on  the  grade  peg,  the 
target  set  and  the  reading  recorded.  A  chainman  holds 
one  end  of  the  tape  or  chain  at  the  grade  peg  and  the  rod- 
man holding  the  other  end  draws  the  tape  taut  and  holds 
the  rod  vertically  for  a  new  peg.  At  each  station  the 
target  is  moved  the  amount  of  fall  per  station.  If  the  line 
is  going  uphill  the  target  is  moved  down  and  it  is  moved 
up  if  the  line  is  going  downhill.  The  leveler  calls  the  cor- 
rect reading  to  the  rodman,  after  the  latter  has  clamped 
the  target,  as  a  check  each  time. 

In  setting  grade  stakes  for  sewers  the  tops  are  placed  at 
some  definite  height  above  the  bottom  of  the  sewer  when 


g6  PRACTICAL  SURVEYING 

the  street  is  already  on  grade.  The  line  being  straight  a 
peg  is  driven  to  grade  at  each  end  of  the  block.  Over  one 
is  set  the  level  and  the  height  from  the  top  of  the  grade 
peg  to  the  center  of  the  telescope  is  carefully  measured. 


^ ^_L A  _A 

y  "N-  t ' 

A 

^^^/P^ 

FIG.  105.     Plunging  a  grade. 


A  light  rod  is  driven  in  the  ground  at  the  other  peg  and 
a  white  card  tacked  to  it  so  the  middle  of  the  card  is  at  a 
height  above  the  top  of  the  peg  equal  to  the  H.  I.  above 
the  other  peg.  The  exact  height  is  measured  and  marked 
on  the  card.  By  manipulating  the  leveling  screws  the 
telescope  is  pointed  to  the  mark  on  the  card.  The  line  of 
sight  is  then  parallel  to  the  grade  line  instead  of  being 
horizontal.  The  target  is  set  on  the  rod  to  a  height  equal 
to  the  height  of  the  cross-hairs  above  the  grade  line.  The 
rod  being  held  vertically  the  bottom  is  on  line  with  the 
tops  of  the  grade  pegs  when  the  horizontal  wire  cuts  the 
center  of  the  target  and  stakes  are  driven  to  grade  wherever 
desired.  This  method  is  termed  "plunging  a  grade." 


FIG.  106.    Sewer  grade  line  on  street. 

When  the  ground  is  irregular  so  that  some  stakes  would 
be  driven  with  tops  below  the  surface  and  others  would 
stand  high  if  driven  to  grade,  it  is  best  to  drive  a  peg  with 
the  top  flush  with  the  ground  at  each  station.  Beside  each 
peg  is  driven  a  stake  marked  with  figures  indicating  the  dis- 
tance to  grade,  a  +  sign  indicating  a  cut  and  a  —  sign  a  fill. 


LEVELING  97 

To  lay  pipes  to  grade,  all  the  witness  stakes  will  be 
marked  +,  for  they  will  all  be  above  grade.  Beside  each 
grade  peg  drive  a  post  and  on  the  opposite  side  of  the 
trench  drive  another.  Measure 

up  from  the  grade  peg  some  dis-  ^4T!'Rod  level 

tance  to  obtain  a  line  a  certain 
number  of  feet  above  the  flow 
line  and  at  this  height  nail  or 
clamp  a  plank  to  the  two  posts, 
the  plank  to  be  perfectly  level. 
This  is  done  at  each  station  and 
a  cord  stretched  from  plank  to 
plank  will  show  the  grade  and 
be  out  of  the  way  of  the  work- 
men in  the  trench.     A  piece  of 
wood     about    one     inch     square     FlG   I0?      Transferring  surface 
with    a   small   bracket  fastened          grade  to  pipe  in  trench, 
at  the  lower  end  is  used  to  ob- 
tain the  grade  at  the  bottom,  the  projecting  arm  resting  in 
the  bottom  of  the  pipe  when  an  upper  mark  touches  the  cord. 
The  rod  must  be  vertical  and  a  rod  level  should  be  used. 

MANUAL   SIGNALS  FOR  SURVEYORS 

It  is  not  always  possible  to  shout  instructions  to  assist- 
ants in  the  field  so  signals  are  necessary.  The  author 
mentioned  his  signals  for  telling  a  rodman  when  to  move  a 
target,  and  the  direction  in  which  to  move  it.  For  con- 
veying information  into  which  numbers  enter  a  very  old 
custom  is  to  write  the  number  with  the  right  hand  on  an 
imaginary  vertical  surface.  When  this  is  done  by  the  rod- 
man the  leveler  reads  the  number  in  a  reversed  direction 
and  the  rodman  reads  it  in  a  reversed  direction  when  the 
writing  is  done  by  the  leveler.  Some  men  in  making  such 
signals  stand  with  their  back  to  the  recipient  and  so  manage 
to  overcome  the  difficulty  of  reading  numbers  reversed. 

In  Engineering  News,  May  22,  1913,  Mr.  F.  T.  Darrow 
illustrated  a  set  of  signals  in  use  on  the  Burlington  Lines, 
these  signals  being  shown  in  Fig.  108.  The  reader  will 
notice  that  one  hand  is  used,  the  other  presumably  holding 
the  rod  or  instrument. 


PRACTICAL  SURVEYING 


6789 

FIG.  108.     Signals  for  surveyors'  assistants. 


Rod  up  or       Move  to  Right   Plumb  the    Set  Hub        Set  onThis 
Give  Line  (or  Left)     RodCRiqht  orqiveTP 

or  Left) 


O.K.  Come  Here      Go  Back     Cant  Get  You    Move  Target 

Up  Cor  Down") 
FIG.  109.     Signals  for  surveyors'  assistants. 


LEVELING  99 

In  Engineering  News,  Aug.  21,  1913,  Mr.  Robert  S. 
Beard  illustrated  a  set  of  signals  requiring  the  use  of  both 
hands.  The  author  has  omitted  from  Fig.  109  the  signals 
for  numbers  as  he  prefers  signals  for  which  one  hand  only 
is  used.  It  is  necessary,  however,  to  convey  other  ideas 
than  numbers  so  those  signals  in  the  article  by  Mr.  Beard 
which  relate  to  common  instructions  are  shown  in  Fig.  109. 
The  signals  may  be  used  by  instrument  men,  rodmen, 
chainmen  or  other  helpers  and  require  no  explanation  as  the 
information  is  placed  under  each  figure.  These  signals  are 
those  in  common  use  by  experienced  men  the  world  over. 

I 


CHAPTER   IV 

COMPASS  SURVEYING 

Nearly  all  land  surveys  before  the  middle  of  the  last 
century  were  made  with  the  compass.  It  is  still  used  to- 
day where  land  is  not  high  in  price. 

About  70  or  80  years  ago  a  fairly  certain  method  was 
developed  for  eliminating  the  effects  of  local  attraction,  but 
it  was  not  in  common  use.  Until  about  100  years  ago 
little  attention  was  paid  to  the  declination  of  the  needle, 
or  variation  as  it  was  commonly  called.  The  declination 
is  the  angle  between  the  true  meridian  and  the  meridian 
indicated  by  the  compass  needle.  The  variation  is  the 
change  in  declination.  In  1836  William  A.  Burt  patented 
the  solar  compass  of  which  he  was  the  inventor  jointly  with 
John  Mullett,  both  Government  surveyors  in  Michigan 
where  local  ore  bodies  attracted  the  needle  and  rendered  it 
valueless  for  running  lines.  With  the  solar  compass  a  true 
north  and  south  line  could  be  run  from  observations  on  the 
sun.  It  is  now  obsolete  so  will  net  be  described  in  this 
book.  The  engineers'  transit  with  solar  attachment  super- 
seded the  solar  compass  and  today  few  men  use  a  solar 
attachment,  direct  observations  on  the  sun  tending  to 
greater  accuracy  and  simplicity. 

After  1836  all  Government  land  surveys  had  to  be  made 
with  a  solar  compass.  About  50  years  later  a  transit  with 
solar  attachment  was  required  and  now  direct  solar  obser- 
vations are  permitted.  The  General  Land  Office  at  present 
prohibits  employees  and  contracting  surveyors  from  de- 
pending to  any  extent  on  courses  derived  from  the  needle. 

The  compass  possesses  the  following  merits: 

1.  It  is  light. 

2.  It  is  readily  set  up  and  therefore  rapid  work  can  be 
done  with  it. 

3.  When  there  is  no  local  attraction  an  error  made  in 

100 


COMPASS   SURVEY  I&C-. 


reading  the  needle  at  any  station  is  not  cumulative,  but  is 
confined  to  the  one  course.  If  a  "back  reading"  is  taken 
the  error  is  caught  and  may  be  corrected. 

4.  Two  points  are  not  required  from  which  to  start,  as 
with  a  transit.     The  needle  pointing  always  in  one  direc- 
tion (when  there  is  no  local  attraction)  any  station  may  be 
used  as  a  starting  point. 

5.  Fewer  helpers  are  needed  than  with  the  transit. 

6.  On  preliminary  lines  and  on  random  lines  considerable 
time  may  be  saved  in  cutting  brush,  for  the  compass  can 
be  set  to  one  side  and  a  parallel  offset  line  run  past  an  ob- 
struction by  merely  setting  the  sight  on  the'  proper  course 
indicated  by  the  needle. 

7.  In  re-  tracing  old   surveys  it  is  better  than  a  more 
exact  instrument  for  the  work  is  duplicated  more  nearly 
when  the  methods  adopted  closely  copy  those  originally 
used. 

8.  When  land  is  very  low  in  value  or  when  the  informa- 
tion wanted  is  only  closely  approximate,  the  compass  is 
the  best  instrument  to  use,  because  men  of  ordinary  ability 
can  do  the  work  and  do  it  quickly. 

Before  mentioning  the  defects  of  the  compass  it  is  neces- 
sary to  say  that  all  land  survey  methods  and  terms  now 
used  were  developed  during  the  era  preceding  the  intro- 
duction of  well-made  accurate  instruments.  The  only 
difference  is  increased  accuracy  so  that  an  understanding 
of  compass  work  is  necessary  for  a  full  knowledge  of  land 
surveying. 

When  the  compass  was  invented  no  one  knows  and  the 
name  of  the  inventor  has  not  been  preserved.  The  in- 
ventor is  supposed  to  have  been  a  Chinese  and  he  lived  at 
least  1900  years  ago.  The  first  mention  in  print  of  the 
compass  is  found  in  a  Chinese  dictionary  printed  in  121 
A.D.,  in  which  book  the  lodestone  is  defined  as  "a  stone 
with  which  an  attraction  can  be  given  to  the  needle." 
The  first  mention  of  the  compass  by  any  European  writerwas 
in  1190  but  it  was  in  common  use  by  seamen  in  1250.  Its 
use  in  surveying  seems  to  be  first  mentioned  about  1300  and 
on  his  voyage  to  America  in  1492  Christopher  Columbus 
verified  the  fact  long  suspected,  that  the  needle  did  not 
point  constantly  to  the  North  Star.  In  1600  Gilbert 


102  PRACTICAL  SURVEYING 

showed  that  the  earth  is  a  great  magnet  with  lines  of  force 
flowing  from  pole  to  pole  in  which  the  needle  settled  and 
that  angels  and  demons  and  the  stars  in  the  heavens 'had 
no  influence  upon  the  needle. 

The  needle  does  not  point  to  the  North  Pole  but  to 
a  shifting  magnetic  pole  located  in  Canadian  territory. 
When  a  compass  is  set  so  that  a  line  drawn  through  the 
north  pole,  the  magnetic  pole  and  the  instrument  will  be 
straight  (that  is  in  the  plane  of  a  great  circle)  we  have  a 
line  of  no  variation,  an  agonic  line. 

At  all  points  east  of  the  agonic  line  the  needle  points  west 
of  true  north  and  has  a  west  declination.  At  all  points 
west  of  the  agonic  line  the  needle  points  east  of  the  true 
north  and  has  an  east  declination.  Since  all  meridians 
pass  through  the  earth's  north  pole  a  true  north  and  south 
line  is  spoken  of  as  "the  meridian." 

This  edition  contains  no  Isogonic  Chart  as  "one  was 
issued  by  the  United  States  Coast  and  Geodetic  Survey  for 
January,  1915,  in  "Special  Publication  on  Terrestrial 
Magnetism,"  which  should  be  owned  by  every  land  sur- 
veyor. A  chart  for  1920  will  appear  about  1922  or  1923  and 
it  is  the  intention  to  issue  such  charts  at  five  year  intervals. 

The  red  lines  show  the  actual  declinations  for  January, 
1915,  as  determined  in  1914  by  5,830  observations  in  North 
America.  The  blue  lines  show  the  annual  change;  the 
north  end  of  the  compass  needle  moving  to  the  westward  at 
all  places  west  of  the  line  of  no  annual  change  and  to  the 
eastward  at  all  places  west  of  that  line. 

To  use  the  Isogonic  Chart. — Find  the  declination  for  191 5  at 
the  point  desired,  as  shown  by  the  figures  in  red.  Add  to  this 
the  amount  of  annual  change  at  that  point  multiplied  by  the 
number  of  years  from  i9i5todate.  The  publication  referred  to 
enables  one  to  trace  changesindeclination  from  the  year  1750. 

Many  old  time  surveyors  knew  little  about  the  decli- 
nation of  the  compass  needle  and  cared  less,  so  their  lines 
do  not  agree  with  the  bearings  set  down  in  the  records. 

The  annual  change  (variation)  of  the  declination  was 
unknown  to  the  majority  of  surveyors  100  years  ago. 

That  minor  variations  in  the  pointing  of  the  needle 
occur  during  the  day  is  not  known  to  all  surveyors  of  the 
present  time. 


COMPASS   SURVEYING  103 

Many  surveyors  paid  small  attention  to  the  probable 
effects  of  local  attraction  and  some  neglected  correcting 
readings  which  were  incorrect  because  of  local  attraction. 

Badly  made  instruments,  some  containing  small  quanti- 
ties of  iron;  bent  pivots;  bent  needles;  neglected  adjust- 
ments; weakly  magnetized  needles;  mistakes  in  reading 
bearings;  bearings  read  only  to  half  degrees  or  quarter 
degrees;  all  these  things  during  the  400  years  during  which 
the  compass  was  the  principal  instrument  used  by  sur- 
veyors finally  resulted  in  its  abandonment  as  an  instru- 
ment of  precision. 

It  is  not  wise  to  purchase  a  second-hand  compass  except 
from  a  reputable  maker,  and  then  only  with  his  certificate 
that  it  has  been  carefully  tested  and  found  to  contain  no 
metal  liable  to  deflect  the  needle.  Cheap  surveying  in- 
struments are  a  poor  investment  and  no  instruments  should 
be  purchased  from  stores  or  agents.  The  needle  of  a  sur- 
veying compass  should  be  at  least  \\  ins.  long  and  deli- 
cately balanced  so  it  appears  to  be  always  quivering  while 
the  ends  remain  in  one  plane.  A  swinging  of  the  ends 
indicates  local  attraction  which  may  be  caused  by  iron  or 
steel  in  the  compass  box,  bodies  of  iron  ore  under  the  sur- 
face of  the  ground,  wire  fences,  wire  in  the  hat  brim  of 
the  instrument  man,  or  steel  or  iron  on  his  person.  When 
a  compass  is  set  up  chains,  tapes,  hatchets  and  all  tools 
containing  metal  which  might  attract  the  needle  should  be 
removed  to  a  considerable  distance. 

Diurnal  variation.  —  In  addition  to  the  secular  variation 
of  the  declination  the  needle  swings  backward  and  for- 
ward each  day  through  an  arc  sometimes  as  great  as  20 
minutes  in  size.  This  variation  cannot  be  predicted  for  it 
alters  with  the  time  of  day,  seasons  of  the  year,  temperature 
and  amount  of  sunshine.  It  is  different  in  localities  not 
far  apart.  In  practical  work  it  is  ignored,  except  in  hot 
weather  when  the  sun  is  bright.  On  such  days  it  is  best 
to  work  only  before  9  A.M.  and  after  3  P.M. 

It  is  impossible  to  make  all  compass  needles  alike.  All 
makers  testify  that  a  number  of  needles  can  be  made  alike 
in  shape  and  of  equal  weight  from  one  piece  of  steel,  yet 
when  magnetized  and  placed  on  pivots  the  readings  will 
differ  as  much  as  10  min. 


104 


PRACTICAL  SURVEYING 


Several  men  reading  the  bearing  shown  by  a  needle  will 
obtain  different  results.  This  personal  error  taken  with 
the  differences  in  needles  shows  that  any  attempt  to  allow 
for  the  diurnal  variation  is  generally  a  needless  refinement. 

Notwithstanding  all  the  drawbacks  mentioned,  a  well- 
trained,  conscientious  surveyor  using  a  well-made  modern 
compass  can  do  very  good  work. 

READING  THE   COMPASS 

Let  the  line  N  ...  5  represent  the  true  meridian  in 
which  the  needle  is  supposed  to  lie  and  the  direction  (bear- 
ing) of  the  line  A   ...  B  is  wanted.     The 
compass  is  set  at  some  point  O  on  the  line 
A   ...  B  and  the  needle  allowed  to  swing 
freely  after  the  instrument  is  leveled.   When 
the  needle  comes  to  rest  the  observer  looks 
through  the  sights  towards  point  B.     The 
compass  circle  is  graduated  from  north  to 
east  and  north  to  west,  the  north  point 
being   marked  O  and   the   east  and  west 
points  90.     It  is  graduated  similarly  from 
FIG   no          south  to  east  and  west,  this  method  be- 
ing termed  "quadrantal  graduation,"  be- 
cause   the    circle    is    divided    into    four    quadrants,    or 
quarters. 


FIG.  in.     Surveyor's  compass  with  open  sights. 

When  the  sights  are  set  on  the  line  A  .  .  .  B  the  needle 
still  points  to  N.     The  zero  being  on  the  line  A  ...  B 


COMPASS   SURVEYING  105 

(in  the  line  of  sight)  the  end  of  the  needle  points  to  some 
number  on  the  circle,  which  number  indicates  the  angle 
N,  0,  B,  between  the  lines  0  ...  TV  and  0  .  .  .  B,  this 
angle  being  called  the  bearing  of  the  line  O-B. 

Fig.  in  illustrates  a  good  type  of  compass  with  variation 
plate  by  means  of  which  the  declination  of  the  needle  is  set 
off.  Underneath  the  compass  is  the  ball-and-socket  joint 
by  means  of  which  the  compass  is  leveled.  Leveling  screws, 
although  a  great  improvement,  are  seldom  used  on  com- 
passes. Two  levels  at  right  angles  show  when  the  plate 
is  level  before  clamping  the  joint.  A  is  the  sight  at  the 
south  end  and  B  the  sight  at  the  north  end  of  the  compass. 
Each  sight  has  a  very  fine  slit,  and  at  the  ends  of  the  slits 
round  openings  are  made  so  objects  may  be  readily  found. 

Sometimes  each  sight  has  two  slits  as  shown.  In  the 
upper  slit  in  one  sight  and  the  lower  slit  in  the  other  sight 
a  fine  wire  or  horse  hair  is  stretched  to  enable  closer  sights 
to  be  taken.  The  surveyor  looks  through  the  plain  slit 
and  bisects  the  object  by  means  of  the  wire  in  the  opposite 
slit. 

The  right  edge  of  B  is  graduated  to  half  degrees  for 
angles  of  elevation  to  be  read  through  the  lower  peephole 
on  sight  A.  The  left  edge  is  graduated  for  angles  of  de- 
pression to  be  read  through  the  upper  peephole  on  sight  A . 
To  read  an  angle  of  elevation  or  depression,  look  through 
the  proper  peephole  at  a  point  the  same  height  above  the 
ground  as  the  compass  plate.  Slide  a  card  up  sight  B 
until  the  top  edge  strikes  the  line  of  sight.  The  card  is 
held  against  the  sight  until  the  angle  is  read. 

At  C  is  shown  a  variation  plate.  By  means  of  a  milled 
screw  the  compass  circle  can  be  revolved  about  the  vertical 
axis  about  30  degrees  each  side  of  the  line  of  sight,  thus 
enabling  the  declination  to  be  set  off  on  the  plate  at  C. 
The  declination  may  be  set  off  in  one  of  three  ways.  If 
a  well-defined  line  can  be  found  of  which  the  bearing  is 
known,  set  the  instrument  on  it  and  sight  to  a  stake  on  the 
line.  By  means  of  the  screw  attached  to  the  variation 
(declination)  plate  move  the  compass  circle  until  the  needle 
points  to  the  course  of  the  line  given  in  the  record.  The 
declination  can  then  be  read  on  the  plate,  which  should  be 
clamped  and  not  again  touched.  Another  method  is  to 


io6 


PRACTICAL  SURVEYING 


lay  off  a  line  in  the  true  meridian,  set  the  compass  on  it  and 
turn  the  compass  ring  until  the  needle  points  to  zero,  when 
the  declination  may  be  read  on  the  plate  at  C.  For  closely 
approximate  work  the  declination  may  be  obtained  from 
the  Isogonic  Chart  and  set  off  on  the  plate. 

The  dial  at  D  is  used  to  keep  tally  in  chaining.     Under 
the  compass  plate  is  a  screw  by  means  of  which  the  needle 

is  raised  and  held  against  the 
glass  cover  when  not  in  use, 
so  it  will  not  dull  or  bend  the 
pivot  when  the  instrument  is 
being  carried. 

On  the  compass  plate  the 
letter  E  is  on  the  left  and  the 
letter  W  is  on  the  right.  The 
needle  points  to  the  north  and 
if  the  line  of  sight  is  pointed 
in  an  easterly  direction  the 
needle  will  show  the  angle, 
as  already  explained,  between 
the  meridian  and  the  line  of 
sight.  The  north  end  of  the 
needle  will  be  to  the  left  of 
the  line  of  sight,  so  by  trans- 
posing the  E  and  W  the 
direction  and  amount  of  di- 
vergence from  the  meridian 
are  both  indicated  by  the 
needle.  The  student  must 
never  forget  that  the  needle 
indicates  the  line  of  sight  but 


FIG.  112. 


Surveyor's  compass  with 


not  jje  jn 

Fig.  112  shows  a  compass 
with  a  telescope  instead  of  open  sights,  a  vertical  circle  for 
reading  angles  of  elevation  and  depression  and  a  level  adapter 
instead  of  ball  and  socket/ 

COMPASS  ADJUSTMENTS 

There  are  five  adjustments  of  the  compass: 
I  .    The  compass  circle  must  be  perpendicular  to  the  vertical 
axis.  —  The  manufacturer  alone  can  make  this  adjustrrent. 


COMPASS   SURVEYING  107 

2.  The  levels  must  be  perpendicular  to  the  line  of  sight.  — 
This  adjustment  cannot  be  made  if  the  compass  circle  is 
not  perpendicular  to  the  vertical  axis,   this  fact  being  a 
check  on   the   first   adjustment.     Level   the   compass   by 
bringing  the  bubbles  to  the  middle  of  their  respective  tubes. 
Turn  the  instrument  180  deg.  and  if  the  bubbles  have  moved 
correct  half  the  difference  by  means  of  the  capstan  head 
screws  holding  the  level  tubes.     Check  and  repeat  until 
the  bubbles  remain  stationary  during  a  complete  revolution 
of  the  compass  on  the  vertical  axis. 

3.  The  needle  must  be  perpendicular  to  the  vertical  axis.  — • 
This  means  the  needle  must  be  horizontal  and  also  straight. 
The  north  end  of  the  needle  tends  to  dip  towards  the  earth 
and  a  coil  of  fine  wire  is  placed  near  the  south  end  to  counter- 
act this  tendency. 

Before  making  any  adjustments  involving  the  needle  it 
should  be  re-magnetized.  Holding  the  needle  in  one  hand 
hold  the  magnetic  pole  of  a  permanent  magnet  firmly  with 
the  other  hand  against  the  needle  near  the  middle  and  pass 
the  magnet  to  the  north  end  of  the  needle.  Before  each 
pass  describe  a  circle  about  one  foot  in  diameter  with  the 
magnet  in  a  plane  with  the  needle.  This  operation  should 
be  repeated  for  the  south  end  of  the  needle  but  care  should 
be  taken  that  the  north  and  the  south  ends  are  applied  to 
the  opposite  poles  of  the  magnet  or  the  work  will  be  wasted. 
About  twenty-five  passes  will  generally  be  sufficient.  This 
operation  should  be  performed  three  or  four  times  each 
year. 

A  weak  needle  is  affected  by  the  friction  on  the  pivot  so 
it  is  necessary  to  keep  it  charged  in  order  to  do  good  work. 
The  method  above  described  is  not  always  satisfactory  but 
in  these  days  of  electrical  power  plants  the  surveyor  can 
readily  charge  his  needle.  Place  the  needle  in  the  magnetic 
field  of  a  dynamo  and  then  test  to  see  if  the  magnetism  is 
reversed.  If  the  needle  points  south  instead  of  north  put 
it  again  in  the  magnetic  field  of  the  dynamo  in  a  reverse 
position  from  that  used  first. 

When  satisfied  that  the  needle  is  properly  charged  level 
the  compass,  bringing  the  north  end  of  the  needle  to  zero 
at  the  north  end.  Read  both  ends  and  reverse  the  instru- 
ment so  the  north  end  of  the  needle  will  read  the  same  as 


108  PRACTICAL  SURVEYING 

the  south  end  of  the  needle  on  the  first  trial.  If  now  the 
south  end  of  the  needle  does  not  read  zero  correct  half 
the  difference  by  bending  the  needle. 

4.  The  point  of  the  pivot  must  lie  in  the  vertical  axis.  — 
Having  performed  the  third  adjustment,  or  made  a  test  that 
showed  the  needle  to  be  straight,  read  the  north  end  of  the 
needle  at  N,  E,  S  and  W  (that  is  at  points  90  deg.  apart). 
Note  the  reading  of  the  south  end  as  each  quadrant  is  read. 
If  in  any  quadrant  the  south  end  passes  over  less  than  90  deg. 
while  the  north  end  passes  90  deg.,  bend  the  pivot  away  from 
that  quadrant.     Test  each  quadrant  in  this  manner  until 
the  needle  swings  freely,  the  two  ends  reading  the  same  in 
each  quadrant  on  points  180  deg.  apart. 

5.  The  line  of  sight  must  lie  in  the  vertical  axis.  —  This 
adjustment  can  be  made  only  by  an  instrument  maker,  but 
the  surveyor  must  be  certain  that  the  slits  in  the  sighting 
vanes  are  vertical.     Sight  at  a  long  plumb  line  and  if  it 
does  not  coincide  with  an  edge  of  the  slit,  put  paper  under 
the  low  side  or  file  the  bottom  of  the  high  side  under  the 
foot  of  the  sight  and  screw  tight.     This  is  to  be  done  with 
both  sights  so  the  edges  of  the  slits  will  be  truly  vertical 
when  the  instrument  is  in  adjustment. 

Caution.  —  Good  work,  even  with  an  instrument  in  per- 
fect adjustment,  cannot  be  done  if  any  unnecessary  walking 
around  the  instrument  is  allowed.  When  the  compass  is 
set  up  and  leveled  it  must  not  be  disturbed. 

CARE   OF   THE   COMPASS 

Treat  the  instrument  with  care  and  it  will  give  good  ser- 
vice for  several  generations.  Avoid  all  shocks  and  jars, 
and  handle  it  so  there  will  be  no  danger  of  bending  any  of 
the  parts.  Carry  it  in  the  hollow  of  the  arm  and  not  as 
one  carries  a  pail  or  basket.  Do  not  use  the  sights  as 
handles. 

Be  careful  of  the  pivot  that  it  will  not  be  dulled  or  bent. 
Lower  the  needle  gently.  When  the  needle  is  let  down  on 
the  pivot  check  the  vibrations  by  lifting  it  off  the  point 
at  each  swing  until  it  settles.  VVhen  moving  to  another 
station  lift  the  needle  before  taking  up  the  compass  and  on 
arriving  at  the  next  station  level  the  compass  carefully, 


COMPASS  SURVEYING 


IOQ 


sight  backward  to  the  station  left  and  let  the  needle  down 
gently.  It  will  then  be  parallel  to  the  position  last  occupied 
and  will  rest  on  the  pivot  without  swinging. 

The  needle  should  always  be  held  off  the  pivot  when  not 
in  use.  When  returned  to  the  case  to  remain  until  again 
required  hold  the  plate  level  and  let  the  needle  down  gently 
on  the  pivot  until  it  swings  in  the  magnetic  meridian. 
Then  raise  it  off  the  pivot. 

Avoid  riding  in  electric  cars  with  a  compass.  If  such 
cars  must  be  used  it  is  well  to  hold  the  compass  as  nearly 
level  as  possible  and  let  the  needle  swing.  Rubbing  the 
glass  with  a  cloth  often  causes  trouble  because  of  f fictional 
electricity,  so  the  surface  should  be  slightly  moistened  by 
breathing  on  it  after  cleaning.  The  compass  needle  should 
not  be  played  with  by  drawing  the  end  from  side  to  side 
with  magnets  or  pieces  of 
metal. 

THE  USE  OF  THE  COMPASS 

\j 

The  field  shown  in  Fig.  113  p 
was  surveyed  and  the  first 
station  occupied  was  at  0, 
the  work  proceeding  with  the 
field  on  the  right  hand  of  the 
surveyor,  "  surveying  with  the 
sun, "  to  use  an  old  expression. 
There  is  no  reason  for  this 
other  than  convenience.  Some 
surveyors  begin  a  survey  at 
the  most  easterly  or  the 
most  westerly  corner  to  keep 
the  signs  either  all  -f  or 
all  —  when  computing  areas. 
When  a  station  other  than 
the  most  easterly  or  westerly 
is  taken  as  a  starting  point 
the  signs  will  be  mixed  and 
unless  the  computer  is  careful  he  may  meet  with  difficulties 
in  making  his  work  close.  It  makes  no  difference  which 
station  is  chosen  from  which  to  start  and  the  lines  may  be 


FIG.  113. 


HO  PRACTICAL  SURVEYING 

run  to  the  right  or  to  the  left.  In  computing  areas  it  is 
not  necessary  to  begin  the  computations  at  the  first  station 
of  the  field  work. 

At  each  station  a  hub  is  set  and  the  instrument  is  set 
over  it  carefully  with  a  plumb-bob  marking  the  center,  if 
a  tripod  is  used.  If  the  compass  is  mounted  on  a  Jacob 
staff  the  staff  is  thrust  into  the  ground  close  beside  the  hub. 
The  compass  is  carefully  leveled  and  the  sight  at  the  N 
end  placed  ahead.  Looking  back  through  the  sights  to 
the  station  last  set  the  needle  is  lowered.  When  it  stops 
swinging  the  reading  is  taken,  thus  checking  the  forward 
reading.  At  station  O  no  backsight  is  taken,  the  instru- 
ment being  placed  so  the  needle,  as  nearly  as  one  can  judge, 
points  to  the  north  before  lowering  it.  The  more  nearly 
the  needle  lies  in  the  meridian  before  being  released  the  more 
quickly  it  comes  to  rest  with  least  wear  on  the  pivot. 

A  back  reading  (backsight)  should  always  be  taken  to 
guard  against  effects  of  local  attraction  and  as  a  check 
on  readings,  for  it  is  easy  to  make  a  mistake  of  10  degrees 
and  sometimes  letters  are  transposed  in  recording.  A  back- 
sight should  give  the  same  reading  as  a  foresight,  that  is, 
the  bearing  should  be  the  same  at  all  points  on  a  line. 

A  reading  is  recorded  by  writing  the  letters  and  the  in- 
cluded angle  as  shown  in  Fig.  113.  With  the  instrument 
at  0  the  forward  reading  was  ^49°  15'  E  to  i.  At  I  the 
bearing  of  the  line  should  be  5  49°  15'  W  to  O.  Beginners 
are  often  told  to  read  the  south  end  of  the  needle  on  back- 
sights. Practical  surveyors  seldom  do  it  for  it  may  lead 
to  mistakes  and  is  often  confusing.  The  common  method 
is  to  set  the  instrument  and  while  the  needle  is  still  clamped 
sight  backward  at  the  last  station.  Then  lower  the  needle 
gently  and  it  will  be  in  the  line  last  read.  When  the  needle 
rests  in  this  bearing  sight  carefully  through  the  slits, 
standing  at  the  N  end  of  the  compass.  Then  go  to  the  S1 
end  and  read  the  needle,  which  should  give  the  same  bear- 
ing as  at  the  last  station.  Then  the  next  bearing  may  be 
read.  This  procedure  gives  a  check  reading  without  mental 
computation  or  any  changing  of  letters.  A  common  ejror 
when  the  south  end  of  the  needle  is  read  is  to  take  86  deg.  to 
be  84  deg.,  83  deg.  to  be  87  deg.,  or  vice  versa  when  the 
reading  is  almost  due  east  or  west;  and,  when  the  reading 


COMPASS   SURVEYING 


III 


is  so  close  to  90°  E  is  often  read  for  Wor  W  for  E.  It  being 
customary  to  complete  a  survey  before  correcting  the  bear- 
ings, such  errors  may  be  serious. 

The  field  is  gone  around  in  the  manner  stated  until 
station  0  is  occupied  a  second  time,  to  obtain  a  backsight 
on  Sta.  5,  if  a  backsight  was  not  taken  to  this  station  at 
the  beginning  of  the  work.  The  notes  are  placed  on  the 
left-hand  page,  the  right-hand  page  being  used  for  sketches 
and  memoranda. 


I 

2 

3 

4 

5 

6 

Station. 

Corrected 
bearing. 

Distance 
(chains)  . 

F.  S. 

B.  S. 

Corr. 

0 

N49°  i5'  E 

7.00 

N5i°  15'  E 

-2°   00' 

N47°  oo'W 

I 

S  46°  15'  E 

8.00 

S  48°  30'  E 

-2°    I5' 

S  45°  45'  E 

2 

S  29°  30'  W 

9.00 

S  30°  oo'  W 

-o°  30' 

S  29°  30'  W 

3 

S  61°  45'W 

5-07 

S  61°  45'W 

±0 

S  61°  45'W 

4 

N22°  30'  E 

6-43 

N22°    30'  E 

±0 

N25°  15'  E 

5 

N4i°  30'  W 

7.00 

N38°  45'  W 

+  2°  45' 

N39°  30'  W 

In  the  field  the  first,  third,  fourth  and  fifth  columns  are 
used.  When  the  survey  is  completed  the  corrections  are 
entered  in  the  sixth  column  and  the  corrected  courses  placed 
in  the  second  column.  The  plus  and  minus  signs  before 
the  corrections  refer  to  the  difference  between  the  corrected 
course  and  the  backsight. 

TO   CORRECT  LOCAL  ATTRACTION 

When  sufficient  care  is  exercised  it  may  be  assumed  that 
all  differences  discovered  between  bearings  from  the  two 
ends  of  a  line  are  due  to  local  attraction.  The  reference 
to  sufficient  care  must  not  be  overlooked,  for  surveyors  often 
make  mistakes  of  several  degrees  in  reading  angles. 

The  following  method  for  correcting  errors  due  to  local 
attraction  has  been  taken  from  Flint's  "  Survey,"  the  first 


112  PRACTICAL  SURVEYING 

edition  of  which  was  published  in  1805.  Sometime  between 
1832  and  1851,  L.  W.  Meech,  A.M.,  was  employed  by  the 
publishers  to  revise  the  work,  which  had  gone  through  six 
editions  before  1835,  and  he  is  credited  with  being  the 
author  of  the  method  since  used  by  a  number  of  men,  but 
mentioned  in  few  texts. 

Examining  the  field  notes  it  is  found  that  the  forward 
reading  from  Sta.  3  agrees  with  the  backsight  from  Sta.  4. 
Evidently  there  was  no  local  attraction  at  either  station  so 
the  forward  and  back  readings  are  assumed  to  be  correct. 
Therefore  the  bearing  of  the  line  from  2  to  3  is  5  29°  30'  W, 
from  3  to  4  is  S  61°  45'  W  and  from  4  to  5  is  N  22°  30'  E. 

Start  at  Sta.  4  to  make  corrections  : 

4.  Correct  bearing  ..........................      #22°  30'  E 

Backsight  ...............................      ^25°  15'  E 

+  2°  45' 

The  backsight  is  greater  so  the  sign  of  the  difference  is 
plus. 

The  sign  of  the  difference  is  minus  when  the  backsight  is 
less  than  the  forward  reading. 

The  difference  is  placed  under  the  course  following.  If 
this  course  and  that  preceding  are  in  the  same  or  opposite 
quadrant  a  plus  sign  becomes  minus  and  a  minus  sign  be- 
comes plus.  The  signs  are  not  altered  when  the  courses 
are  not  in  the  same  or  an  opposite  quadrant.  When  the 
proper  sign  is  determined  the  correction  is  applied. 

5.  Forward  reading  .........................      ^38°  45',  W 

Correction  ..............................      +   2°  45' 

30'  W 


The  correction  is  added  here,  for  courses  4  and  5  are  in 
adjacent  quadrants  and  the  backsight  was  greater  than  the 
forward  bearing. 

5.    Corrected  bearing  ........................      N  41°  3°'  w 

Backsight  ...............................      ^39°  30'  W 

—     2°    00' 

o.    Forward  reading  ..........................      N  51°  15'  £ 

Correction  ...............................      -   2°  oo' 

N  49°  15'  E 


COMPASS   SURVEYING  113 

The  correction  is  subtracted  as  the  backsight  was  less 
than  the  forward  bearing  and  the  two  bearings  are  in  ad- 
jacent quadrants. 


o. 


i. 


Corrected  bearing ^49°  15'  E 

Backsight ^47°  oo'  E 

-    2°    15' 

Forward  reading S  48°  30'  E 


Correction 


-   2°  15' 


5  46°  15'  E 

I.   Corrected  bearing. 5  46°  15'  E 

Backsight 5  45°  45'  E 

-  o°  30' 

Forward  reading 5  30°  oo'  W 

Correction -   o°  30' 


2. 


2. 


5  29°  30'  W 

Corrected  bearing S  29°  30'  W 

Backsight 5  29°  30'  W 


If  the  last  corrected  bearing  checks  with  its  backsight 
the  courses  may  be  assumed  to  be  correct,  but  such  a  close 
check  is  not  always  obtained. 

This  method  also  takes  care  of  the  diurnal  declination  of 
the  needle  and  all  errors  are  greatly  reduced  even  when  not 
entirely  eliminated. 

In  the  following  example  the  first  course  is  assumed  to 
be  correct.  The  difference  between  the  forward  and  back 
readings  gives  a  correction  of  +3  deg.  but  courses  I  and  2 
being  in  the  same  quadrant  the  sign  is  changed. 


Station. 

Bearing. 

Distance, 
chains. 

P.  S. 

B.S. 

Corr. 

0 

N85°W 

120.00 

N85°W 

±0 

88° 

I 

Ni4°W 

60.00 

Ni7°W 

10 

-3° 

2 

N75*°E 

90.00 

N74°E 

I5a 

+  i*° 

73* 

3 

S  27^°  E  . 

102.60 

S  29*E 

-2° 

27* 

114  PRACTICAL  SURVEYING 

Correct  the  following  field  notes: 


Station. 

F.  S. 

B.  S. 

Station. 

F.  S. 

B.  S. 

I 

Ni9°E 

18° 

I 

S  25°   W 

25° 

2 

S  78°  E 

79° 

2 

S  10°   W 

w|' 

3 

4 

S  29°  E 
S  52|°W 

26r 

53° 

3 
4 

S  7Si°W 
Nn°   E 

76° 

5 

S  i4i°E 

is*0 

5 

N    2°   E 

si° 

6 

N8si°E 

8Si° 

In  correcting  bearings  by  the  foregoing  method  a  check 
will  not  always  be  obtained  because  needle  readings  are 
taken  to  the  nearest  quarter  degree  with  the  best  com- 
passes. Closer  readings  are  only  estimates.  The  correc- 
tion for  local  attraction  will  be  found  generally  to  be  more 
accurate  than  the  reading  of  bearings. 

All  compass  surveys  should  be  corrected  for  the  effects 
of  local  attraction  and  thus  the  greatest  source  of  error  is 
guarded  against.  When  several  lines  seem  to  show  an 
agreement,  as  in  the  worked  example,  it  is  taken  for  granted 
the  true  bearings  were  obtained,  although  the  local  attrac- 
tion may  have  been  the  same  in  amount  and  direction  at 
certain  stations,  in  which  case  only  the  average  probable 
bearings  were  obtained.  If  local  attraction  is  proved  to  be 
present  at  all  stations  then  that  course  where  it  seems  to 
be  least  may  be  taken  as  the  standard. 

When  the  object  of  the  survey  is  to  obtain  the  area  of  a  piece 
of  land  the  corrected  bearings  will  give  accurate  results  no 
matter  how  far  wrong  the  selected  initial  bearing  may  be. 
The  effect  of  local  attraction  being  eliminated  the  correct  angles 
between  the  lines  are  obtained. 

When  the  survey  is  to  be  recorded  and  future  surveyors 
are  to  be  guided  by  the  record  it  will  be  necessary  to  have 
the  true  bearings. 

Set  two  stakes  on  the  true  meridian,  at  some  place  not 
far  from  the  field,  where  it  is  known  that  no  local  attrac- 
tion exists.  The  compass  is  set  over  the  stake  at  the  south 
end  of  the  line  and  sighted  to  the  stake  at  the  north  end. 
By  means  of  the  declination  plate  screw  the  compass  box 
is  revolved  on  the  vertical  axis  until  the  needle  reads  zero 
and  the  declination  is  set  off  on  the  plate.  A  sight  is  then 


COMPASS   SURVEYING 


taken  to  some  corner  on  the  survey  and  the  bearing  is  read. 
The  line  is  then  run  perfectly  straight  by  backsights  and 
foresights  without  using  the  needle,  until  the  corner  is 
reached.  Setting  up  on  the  corner  a  backsight  is  taken 
along  the  line  just  run  (the  starting  point  of  which  is  within 
an  area  having  no  local  attraction)  and  the  compass  box  is 
turned  until  the  needle  points  to  the  correct  bearing.  This 
of  course  alters  the  declination,  which  makes  no  difference 
as  it  is  not  necessary  to  record  the  declination  when  a  survey 
is  stated  to  have  been  run  from  a  true  meridian.  Having 
set  the  needle  the  first  forward  reading  will  be  correct  and 
all  other  bearings  may  be  corrected  by  using  it  as  a  standard. 

THE   TRUE   MERIDIAN 

The  declination  of  the  needle  may  be  set  off  on  the  com- 
pass plate  after  using  the  Isogonic  Chart  as  already  de- 
scribed. If  this  method  is  not  available  the  true  meridian 
may  be  obtained  with  an  accuracy  equal  to  that  of  good 
compass  work,  from  a  record  of  equal  altitudes  of  the  sun. 

On  a  level  area  set  a  straight  pole 
vertically.  About  three  hours  before 
noon  drive  a  small  stake  in  the  ground 
at  the  end  of  the  shadow  of  the  pole. 
With  a  chain  or  tape  attached  to  a 
ring  around  the  pole  strike  a  semi- 
circle with  a  radius  equal  to  the 
length  of  the  shadow.  If  a  cord  is 
used  a  strong  pull  may  stretch  it  and  a 
wire  will  do  instead  of  a  chain  or  tape. 

In  the  afternoon  drive  a  small  stake  on  the  end  of  the 
shadow  when  it  touches  the  circumference  of  the  circle. 
Midway  between  the  two  stakes  make  a  mark.  A  line 
through  this  mark  and  the  center  of  the  pole  will  define 
the  true  meridian,  which  may  be  laid  off  to  any  length  by 
using  a  chalk  line  stretched  from  the  pole  over  the  meridian 

point. 

MAKING  A  COMPASS  SURVEY 

Field  notes  should  be  clear  and  the  surveyor  must  never 
forget  that  other  men  may  have  to  use  his  notes  after  he  is 
dead.  Nothing  essential  should  be  omitted,  nothing  non- 
essential  should  be  put  down. 


vertical  Polo 


FIG.  114.  True  meridian 
from  equal  altitudes  of 
sun. 


Il6  PRACTICAL  SURVEYING 

Common  custom  indicates  that  for  compass  work  as 
well  as  transit  work  it  is  most  convenient  to  use  the  left- 
hand  page  of  the  book  for  notes  and  the  right-hand  page  for 
sketches.  The  stations  should  begin  with  0  and  at  the 
bottom  of  the  page.  One  line  should  be  used  for  each 
chain  or  on  long  lines  every  tenth  chain,  even  when  no 
stakes  are  set  between  instrument  points,  or  stations.  In 
this  way  fences,  roads,  buildings,  etc.,  may  be  located  by 
measurement  and  be  shown  by  sketches  in  their  relative 
positions.  See  Fig.  7. 

From  each  instrument  point  bearings  to  objects  are 
often  taken  and  measurements  made  and  the  surveyor 
should  carry  a  thin  flexible  ruler  in  his  field  book  for  sketch- 
ing purposes,  drawing  lines  as  nearly  as  possible  in  the 
right  direction. 

A  good  workman  is  judged  by  the  evidences  of  his  work, 
and  neat  full  notes  go  far  towards  fixing  the  reputation  of 
a  surveyor,  even  when  the  notes  may  fall  into  the  hands  of 
persons  ignorant  of  surveying.  The  criterion  by  which  the 
quality  of  field  notes  may  be  judged  is  that  they  may  be 
sent  to  a  draftsman  so  he  can  map  the  survey  and  no  further 


PIG.  115. 

information  is  required  by  him.  When  an  instrument 
man  must  plat  his  own  notes  or  stay  near  a  draftsman 
who  is  using  them,  he  is  incompetent. 

It  is  seldom  possible  to  place  an  instrument  on  a  corner, 
for  fences  or  hedges  usually  cover  the  lines,  so  surveys  are 
made  on  offset  lines. 

Fig.  115  illustrates  a  method  used  for  offsetting  when 
using  a  compass.  The  point  D  is  selected  by  eye  to  bisect 
the  included  angle  between  AB  and  BC.  The  instrument 
is  set  over  D  but  the  offset  line  parallel  to  AB  is  measured 


COMPASS   SURVEYING  117 

to  d' ',  this  point  being,  as  nearly  as  may  be  estimated,  per- 
pendicular to  AB  opposite  B.  Similarly  the  offset  line 
parallel  to  B C  is  measured  from  d. 

When  boundaries  are  crooked  one  line  may  be  run.  At 
each  angle  the  perpendicular  distance  is  measured  from 
the  offset  line  and  recorded.  Sometimes  the  surveyed 
line  is  run  on  one  side  entirely,  so  all  measurements  will  be 


FIG.  116. 


to  the  right  or  to  the  left,  instrument  men  occasionally 
forgetting  to  set  down  the  right  letter.  To  prevent  errors 
sketches  should  be  made  to  supplement  the  written 
notes. 

When  buildings,  fences  or  other  objects  are  to  be  located 
for  the  purpose  of  showing  them  on  a  map,  bearings  are  read 
to  them  and  the  distances  measured. 

Sometimes  the  surveyor  surveys  an  interior  polygon, 
from  the  corners  of  which  bearings  are  taken  to  the  corners 
of  the  field  and  the  distances  measured  as  shown  in  Fig.  117, 
where  the  instrument  was  set  only  at 
the  corners  A,  B,  C,  D  and  E  with  bear- 
ings and  distances  taken  to  the  corners 
o,  i,  ...  8  of  the  field.  This  saves 
time  in  the  field,  when  the  surveyor  and 
his  assistants  are  under  pay,  and  in- 
creases  the  time  in  the  office  when  the 
draftsman  alone  is  working.  The  possibility  of  mistakes 
is  increased  because  of  the  greater  number  of  compu- 
tations and  lack  of  checks.  A  good  surveyor  never  neg- 
lects to  check  every  operation  when  possible. 

The  office  work  in  the  case  of  a  field  surveyed  by  radiating 
courses  from  an  interior  polygon  begins  by  making  correc- 
tions for  local  attraction  at  each  station  and  then  correct- 
ing the  radiating  bearings.  This  method  of  surveying  by 
radiation  is  very  old. 


Il8  PRACTICAL  SURVEYING 


ANGLES  FROM  BEARINGS 

When  much  local  attraction  exists  it  is  a  good  plan  to 
mark  on  the  map  the  angles  at  the  corners  after  correcting 
the  bearings.  A  note  should  call  attention  to  the  fact  that 
these  angles  are  to  be  used  in  preference  to  the  bearings  on 
re-surveys,  for  the  cause  of  the  local  attraction  may  be 
removed  after  the  original  survey. 

The  angles  may  be  deflections  or  included  angles. 

An  angle  is  the  amount  of  divergence  between  two  inter- 
secting lines  in  a  plane. 

A  deflection  or  exterior  angle  is  the  difference  in  direc- 
tion between  two  courses,  shown  by  d  in  Fig.  118. 


FIG.  118.     Interior  and  deflection  angles.  FIG.  119. 

An  included,  or  interior,  angle  is  the  supplement  of  the 
exterior  angle,  that  is  180°  —  d.  In  Fig.  118  the  letter  i 
is  used  to  indicate  the  included  angles. 


RULES  FOR  OBTAINING  DEFLECTION  ANGLES     ' 

I.    First  letters  alike  and  last  letters  alike.     Fig.  119. 

Rule.  —  The  deflection  is  equal  to  the  difference  between 
the  courses. 

The  deflection  is  to  the  right  in  the  TV  E  and  5  W  quad- 
rants when  the  following  course  is  the  greater. 

^70°^: 

N20°  E 

50°  deflection.  If  course  N  70°  E  follows  N  20°  E 
the  deflection  is  50°  R.  If  N  20°  E  follows  N  70°  E  the 
deflection  is  50°  L. 


COMPASS  SURVEYING 


IIQ 


The  deflection  is  to  the  left  in  the  S  E  and  N  W quadrants 
when  the  following  course  is  the  greater. 

N  70°  W 

N20°W 

50°  deflection.  If  course  N  70°  W  follows  N  20°  W 
the  deflection  is  50°  L.  If  N  20°  W  follows  N  70°  W  the 
deflection  is  50°  R. 

(Note.  —  N  70°  E  and  5  70°  W  describe  the  same  line 
viewed  from  different  ends.) 


FIG.  120. 


FIG.  121. 


2.    /<Yr.y/  letters  alike  and  last  letters  unlike.     Fig.  120.  ** 

Rule.  —  Add  the  courses. 

Bearing  changing  from  W  to  E,  going  north  the  deflec- 
tion is  to  the  right;  going  south  the  deflection  is  to  the  left. 

Bearing  changing  from  E  to  W,  going  north  the  deflection 
is  to  the  left;  going  south  the  deflection  is  to  the  right. 


-V 


B 


tb) 


FIG.  122. 


3.    First  letters  unlike  and  last  letters  alike.     Fig.  12 1. 

Rule.  —  Subtract  the  sum  of  the  courses  from  1 80°.  "" 

Going  east  the  deflection  is  to  the  right  when  N  changes 
to  5,  and  to  the  left  when  S  changes  to  N. 

Going  west  the  deflection  is  to  the  right  when  S  changes 
to  N  and  to  the  left  when  ^V  changes  to  S. 


120  PRACTICAL  SURVEYING 

4.  First  letters  unlike  and  last  letters  unlike.     Fig.  122. 
Rule. — Subtract  the  difference  of  the  courses  from  180  deg. 
Going  north  if  the  bearing  changes  toward  the  west,  or 

going  south  the  bearing  changes  toward  the  east  the  de- 
flection is  left. 

Going  north  if  the  bearing  changes  toward  the  east,  or 
going  south  the  bearing  changes  toward  the  west  the  de- 
flection is  right. 

RULES  FOR  OBTAINING  INCLUDED   ANGLES 

5.  If  the  first  letters  are  alike  and  the  last  are  alike  sub- 
tract the  greater  course  from  180  deg.  and  add  the  smaller 
course.     Fig.  123. 

6.  If  the  first  letters  are  alike  and  the  last  are  unlike 
subtract  the  sum  of  the  courses  from  1 80  deg.     Fig.  124. 

V 

I 

I 
I 


S 

FIG.  125.  FIG.  126. 

7.  If  the  first  letters  are  unlike  and  the  last  are  alike, 
add  the  courses.     Fig.  125. 

8.  If  the  first  letters  are  unlike  and  the  last  are  unlike 
the  included  angle  is  equal  to  the  difference  of  the  courses. 
Fig.  126. 

It  will  be  seen  that  the  required  angle  in  each  case  is  ^4,  B, 
C  in  the  last  four  rules. 

The  student  should  have  considerable  exercise  in  obtain- 
ing angles  from  bearings.  This  can  be  done  very  well  on 
a  table  by  using  a  circular  protractor.  Procure  a  paper 
protractor  8  ins.  in  diameter  and  trim  it  to  circle.  Number 
the  graduations  each  way  from  a  meridian  line  from  o  deg.  to 
90  deg. ,  lettering  the  quadrants  like  a  compass,  with  E  and  W 
reversed.  Tack  down  a  sheet  of  drawing  paper  and  on  it 
draw  a  straight  line  to  represent  the  true  meridian.  Lay  the 
protractor  on  this  line  so  the  zero  points  touch  it.  Draw  a 


COMPASS    SURVEYING 


121 


line  normal  to  the  meridian  and  set  the  center  of  the  pro- 
tractor at  the  intersection  of  the  two  lines. 


FIG.  127.     Obtaining  angles  from  bearings. 

To  plot  a  course  turn  the  protractor  until  the  required 
angle  touches  the  meridian  line.  On  the  drawing  paper 
mark  a  dot  at  the  N  zero  point  and  opposite  it  write  the 


122  PRACTICAL  SURVEYING 

course.     Set  off  the  second  course.     Then  with  the  N  zero 
at  one  course  the  included  angle,  or  the  deflection  angle, 
can  be  read  off  as  a  check  on  the  calculations. 
To  obtain  the  true  bearing  from  a  random  line. 

In  Fig.  128  two  cases  are  shown 
of  random  lines.     In  (a)  the  dotted 
line  represents  a  boundary  and  the 
surveyor  started  from  A  to  re-trace 
the  line  but  when  the  required  dis- 
tance was  measured  found  himself 
at  BI  instead  of  B.     The  difference 
d  was  then  measured. 
In  (b)  the  case  is  similar  but  an  offset  line  A\B\  was  run 
so  that  instead  of  the  offset  at  each  end  being  0,  it  was  O 
at  Ai  and  O1  at  B\. 

The  difference      d  =  0'  -  0. 

Then  angle  a  =  *™**  =  *™  *  d. 

AtXBi       AXBi 

Assume  the  length  of  the  line  to  be  20  chains  and  d  =  20 
links  =  0.20  chain. 


0-573  X  60  =  34.38  min. 

The  line  AB\  (or  A\B\)  was  run  on  a  bearing  of  N  45° 
oo'  E,  but  as  it  ran  to  the  right  of  the  true  line  the  bearing 
must  be  corrected  to  the  left  and  becomes  N  44°  25'  E. 

Suppose  the  original  field  notes  to  have  called  for  a  bear- 
ing of  N  45°  R  for  line  AB  and  the  re-tracement  as  above 
showed  a  difference  of  35  min.  between  the  actual  bear- 
ing and  that  given  in  the  old  field  notes  ;  this  difference  can 
be  set  off  on  the  variation  plate.  The  whole  survey  can 
then  be  re-traced  according  to  the  recorded  bearings,  pro- 
vided the  original  surveyor  made  no  serious  errors  in  read- 
ing the  needle. 

MAKING  THE  MAP 

In  the  center  of  a  sheet  of  paper  rule  two  long  straight 
lines  perpendicular  to  each  other.  The  line  from  the 
bottom  to  the  top  is  the  true  meridian  and  the  line  from  left 
to  right  is  an  east  and  west  line. 


COMPASS   SURVEYING 


123 


Graduate  a  paper  protractor  in  quadrants  but  do  not 
transpose  the  E  and  W  as  on  a  compass  plate.  On  the 
protractor,  which  should  be  14  ins.  in  diameter,  draw  a 
fine  ink  line  joining  the  N  and  S  zeros  and  another  joining 
the  E  and  W,  90°,  points.  At  the  center  where  the  lines 


FIG.  129.    Paper  protractor  for  plotting  angles. 

intersect  and  at  a  couple  of  places  on  each  line  cut  square 
or  round  holes  about  \  in.  across,  with  clean  smooth  edges. 
Trim  the  protractor  along  the  edge  into  circular  form,  for 
it  is  printed  on  a  square  sheet. 

Lay  the  protractor  on  the  drawing  paper  over  the  two 
ruled  lines  so  they  may  be  seen  through  the  holes  and  their 
intersection  will  indicate  the  exact  center  of  the  protractor. 


124 


PRACTICAL  SURVEYING 


To  keep  it  in  position  two  thumb  tacks  may  be  used,  or, 
if  the  holes  will  be  considered  objectionable  in  the  map 
use  paper  weights.  Very  good  paper  weights  are  made 
of  chamois  skin  bags  containing  two  Ibs.  of  fine  shot,  an 
outer  bag  of  cloth  being  used  as  a  cover. 

The  protractor  being  carefully  set,  with  a  sharp-pointed 
pencil  mark  all  the  angles  on  the  drawing  paper,  numbering 
each  dot  to  correspond  with  the  number  of  the  course,  or 
write  the  course  after  each  dot.  When  all  the  angles  are 
marked  remove  the  protractor  and  connect  by  fine  lines 
each  dot  with  the  center.  The  result  will  look  like  Fig.  130. 


FIG.  130.    Plotting  bearings.  FIG.  131.    Plat  made  by  bearings. 

Select  a  starting  point  and  with  triangles,  or  with  trian- 
gle and  straight-edge,  transfer  the  first  course  to  that  point 
and  draw  a  line  parallel  to  the  course.  With  the  scale  set 
off  the  length  on  the  line  and  mark  the  end  for  Sta.  I. 
From  this  station  set  off  the  second  course  and  proceed  in 
this  manner  until  the  boundary  is  all  platted.  The  com- 
pleted result  is  shown  in  Fig.  131. 

On  long  lines  of  survey  the  true  meridian  is  carefully 
transferred  by  straight-edge  and  triangle  to  a  convenient 
place  so  it  will  not  be  necessary  to  transfer  the  courses 
farther  than  is  convenient  with  a  triangle,  thus  avoiding 
danger  of  slipping  and  consequent  error. 

PLOTTING  BY  DEFLECTIONS 

Some  inexperienced,  or  poorly  instructed,  men  will  use 
small  protractors  and  plot  deflections.  Each  course  is 
laid  off  to  a  considerable*  length  and  the  distance  marked. 


COMPASS   SURVEYING 


125 


The  protractor  is  placed  with  center  on  the  station  and  the 
deflection  angle  set  off.  A  line  is  then  drawn  through  the 
two  points,  the  length  of  the  next  course  marked  and  the 


FIG.  132.     Platting  deflections. 

operation  repeated  until  all  the  courses  are  plotted.  The 
method  is  troublesome  and  very  inaccurate  for  the  errors 
are  cumulative. 

Another  method  is  to  use  a  heavy 
steel  straight-edge  and  large  triangle 
and  carry  the  meridian  through  the 
end  of  each  course.  A  large  pro- 
tractor is  placed  on  each  station  and 
the  following  course  laid  off.  This 
is  an  improvement  on  the  method  of  plotting  deflections, 
but  slow  and  not  so  accurate  as  the  method  first  described. 

Other  methods  of  plotting,  requiring  some  knowledge  of 
Trigonometry,  are  described  in  Chapter  VI. 

DISTRIBUTING  ERRORS 

Errors  in  surveys  show  up  when  making  the  map,  for 
the  last  distance  will  carry  the  end  past  the  last  station 
(the  starting  point),  or  perhaps  fall  short.  It  will  also  be 


FIG.  133.     Platting  from 
meridians. 


126 


PRACTICAL  SURVEYING 


to  one  side.     A  plat  of  a  survey  is  apt  to  look  like  Fig. 

134. 

Measure  the  distance  from  0'  to  o  and  call  it  a.       Call 

the  sum  of  all  the  courses  b.     The  closing  error  =     ,— •   • 

Vtf/b 

If  this  exceeds  5£<y,  or  any  limit 
of  error  selected,  then  each  course 
should  be  mentally  examined  to 
discover  if  possible  whether  a  mis- 
take was  made.  If  no  evidence  is 
found  it  will  be  best  to  re-run  the 
lines  and  bring  the  error  within 
proper  limits. 

An  error  within  proper  limits  may 
be  distributed  over  the  courses  in 
proportion  to  the  actual  or  weighted  lengths. 

To  weight  errors  the  survey  is  mentally  reviewed  and 
those  courses  on  which  no  difficulties  were  encountered  are 
given  a  weight  of  I,  those  on  marshy  land,  1.5,  those  on 
steep  land,  2,  these  values  being  here  chosen  merely  for 
illustrative  purposes.  The  values  indicate  the  relative 
probability  of  errors  occurring. 

v.  Multiply  each  course  by  the  weight  assigned  to  it  and 
add  the  new  lengths  progressively.     This  new  total  length 


FIG.  134.     Plat  in  which 
error  is  not  distributed. 


14  '8  21  18  7-5        6        6.       6.5 

FIG.  135.     Graphical  method  for  distributing  errors. 


is  measured  off  on  a  straight  line  and  at  one  end  the  total 
error  is  measured  off  on  a  perpendicular  line  and  a  triangle 
formed.  From  each  station  point  a  perpendicular  is 
erected  to  intersect  the  hypothenuse  and  the  length  of  the 
perpendicular  at  any  station  represents  the  amount  of 


error. 


A  line  is  drawn  from  O'  to  O  on  the  map  and  parallel 
lines  are  transferred  to  each  station  as  shown  in  Fig.  136. 
The  ends  are  connected  and  the  survey  is  closed. 


COMPASS   SURVEYING 


127 


Multiplying  a  distance  by  the  weight  assigned  lengthens 
it  and  thus  gives  it  that  much  greater  proportion  of  error. 
This  is  evident  from  the  law  of  pro- 
portionality of  triangles.  Instead  of 
making  a  triangle  the  error  may  be 
distributed  as  follows: 

Divide  the  error  by  the  total  length 
of  all  the  courses,  after  weighting  same. 
Multiply  the  quotient  by  each  course 
to  obtain  the  error  for  each  course. 

Example.  -  -  Closing  error  0.19 
chain. 


FIG.    136.      Plat  of  field 
with  errors  distributed. 


No.  of  course. 

Distance, 
chains. 

'Weight. 

Weighted 
distance. 

Error. 

O 

4-75 

I 

4-75 

O.OlS 

I 

3-lS 

I 

3-15 

0.012 

2 

6.22 

I 

6.22 

0.024 

3 

7.10 

i-S 

10.65 

0.041 

4 

9-05 

2 

I8.IO 

0.069 

5 

2.76 

2-5 

6.90 

O.O26 

49-77 

O.igo 

0.19 
4977 


=  0.0038  +  per  chain. 


When  the  lines  are  adjusted  and  the  survey  is  closed  the 
map  may  be  completed.  From  the  notes  and  sketches  in 
the  field  book  plat  the  fences,  etc.  Draw  meridians  through 


FIG.  137. 

stations  from  which  bearings  were  taken  to  objects;  place 
a  protractor  on  the  meridians  and  lay  off  the  bearings. 
For  this  work  a  transparent  protractor  with  scale  on  edge 
is  very  convenient.  It  is  also  a  useful  tool  to  carry  in  the 


128  PRACTICAL  SURVEYING 

field  book  to  assist  in  making  sketches.  The  center  is 
placed  on  the  station  and  the  meridian  passes  through  the 
angle.  The  edge  then  indicates  the  bearing  to  the  object, 
the  distance  to  which  is  measured  by  means  of  the  gradu- 
ations. 

Protractors  are  made  of  horn,  paper,  metal,  celluloid, 
ivory  and  rubber.  The  prices  range  from  fifteen  cents  to 
eighty-five  dollars.  For  ninety  per  cent  of  the  work  done 
by  surveyors  and  engineers,  I4~in.  paper  protractors  grad- 
uated to  J  degrees  are  better  than  the  more  expensive 
kinds. 

To  find  the  area  of  a  surveyed  field,  first  plot  it  as  de- 
scribed and  when  the  lines  are  closed  divide  it  into  quad- 
rilaterals, triangles,  etc.;  obtain  the  area  of  each,  and  the 
sum  of  these  areas  is  the  area  of  the  field. 

OBTAINING   AREAS   BY  COMPUTATION   WITHOUT  MAKING 

A  MAP 

All  compass  bearings  are  angles  measured  to  the  right  or 
left  of  the  true  meridian. 

A  course  may  therefore  be  drawn,  forming  the  hypothe- 
nuse  of  a  right-angled  triangle  the  base  of 
which  is  on  the  meridian.  The  base  is  called 
the  latitude,  for  latitude  is  measured  north 
and  south  from  the  equator.  The  alti- 
tude is  called  the  departure,  for  it  measures 
the  amount  by  which  the  course  "departs" 
from  a  true  north  and  south  line. 

By  some  writers  within  recent  years,  the 
"departure"   has   been   termed    "longitude 
difference."     The  reasoning  is  that  through 
each  point  a  true  meridian  may  be  drawn 
and    longitude   being    measured   on    circles 
normal  to  meridians  the  distance  between 
F         g          meridians  is  the  difference  in  longitude.     A 
Traverse  Table  (page  142),  used  for  compass 
surveys,  contains  the  latitude  and  departure  for  all  bearings 
from  o°  to  90°,  varying  by  quarters  of  a  degree. 

Example  I.  —  Find  the  latitude  and  departure  for 
N  5i°  W  23.77  chains. 


COMPASS   SURVEYING 


129 


Looking  in  the  column  headed  Course,  trace  down  to  the 


horizontal   line 
follows : 


opposite   5°  15'.     Arrange  in   columns  as 


Latitude. 

Depart  tire. 

Dist.  2X10  = 
Dist.  3X1   = 
Dist.  7X0.1  = 
Dist.  7X.oi  = 

19.916 
2.9874 
0.6970 
0.0697 

1-83 
0-2745 
o  .  0645 
0.0064 

23.6701 

2.I7S4 

Therefore    in    going    AT  5°  15'  W  23.77   chains   the   line 
runs  north  23.67  chains  and  west  2.18  chains. 

For  an  angle  of  45°  the  latitude  and  departure  are  equal. 

For  an  angle  of  o°  the  latitude  is  I  and  the  departure  =  o. 

For  an  angle  of  90°  the  latitude  is  zero  and  the  departure 
=  i. 

Therefore  the  latitude  of  an  angle  less  than  45°  is  equal 
to  the  departure  of  an  angle  of  the  same  size  between  45° 
and  90°.  For  example  the  latitude  for  13°  is  the  depar- 
ture for  77°  and  vice  versa.  This  fact  saves  much  labor 
in  the  computation  of  tables  and  economizes  space  in  the 
printing  of  tables.  For  all  angles  under  45°  read  the  courses 
on  the  left-hand  edge  of  the  page  going  down,  using  the 
Lat.  and  Dep.  columns  at  the  top.  For 
all  angles  over  45°  read  the  courses  on 
the  right-hand  edge  of  the  page  going 
up,  using  the  Lat.  and  Dep.  columns  at 
the  bottom. 

North  latitude  is  positive  (  +  )  and 
south  latitude  is  negative  ( — ) .  Similarly 
east  departure  is  positive  (  +  )and  west 
departure  negative  (  — ).  This  is  illus- 
trated in  Fig.  139.  All  distances  due 
north  and  due  east  are  said  to  be  meas- 


FIG.  139. 


ured  in  a  positive  sense  and  all  distances  due  south  and  due 
west,  the  opposite  of  north  and  east  respectively,  are  said 
to  be  measured  in  a  negative  sense.  If  a  man  starts  from 
a  certain  point  and  measures  north  +10  chains,  then  meas- 
ures back  on  the  line  south  —4  chains,  he  ends  at  a  point 
+  io  —  4  =  +6  chains  north  of  the  starting  point. 


130  PRACTICAL  SURVEYING 

Example.  —  A  line  is  surveyed  as  follows : 

N  50°  E,  1 1  chains, 
N  7°  W,  12  chains, 
,S  14°  E,  22  chains. 

How  far  apart  are  the  two  ends  of  the  line,  measured  by 
the  latitude  and  departure  of  a  line  connecting  the  points? 


Bearing. 

Distance. 

N+ 

S- 

E+ 

W- 

N    s°E 

II 

10.958 

O.CKQ 

N    7°W 

12 

II   911 

I      4.62 

S  i4°E 

22 

21-347 

0-532 

22.869 

21-347 

I-49I 

1  .462 

After  ruling  a  sheet  of  paper  as  above  and  filling  in  the 
bearings  and  distances  use  the  traverse  table  and  place  the 
results  in  the  proper  columns. 


NS°E. 

N7°W. 

S  14°  E. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

Dist. 

Lat. 

Dep. 

IO 

I 

9-96I9 
0.9962 

0.8716 
0.0872 

IO 

2 

9-9255 
1-9851 

1.2187 
0.2437 

20 
2 

19.406 
I-94I 

0.4838 
0  .  0484 

10.9581 

0.8588 

II  .9106 

1.4624 

21-347 

°-5323 

North  (  +  )  latitudes 
South  (-)  latitudes 

East     (+)  departures 
West    (  — )  departures 


22  .  869 
21.347 
1.522  Diff.  (+) 


i  .  462 

0.029    Diff.  (+) 


The  last  point  set  is  1.522  chains  north  and  0.029  chain 
east  from  the  starting  point. 

In  computing  areas  of  compass  surveys  the  process 
commonly  followed  is  essentially  that  of  reducing  the  figure 
to  a  triangle  of  equivalent  area  and  then  finding  the  area 
of  the  triangle.  This  is  the  method  of  Double  Meridian 


COMPASS   SURVEYING  131 

Distances.  In  some  old  American  textbooks  it  was  known 
as  the  Pennsylvania  Method  but  the  original  discoverer 
was  Thomas  Burgh  of  Ireland  in  1778,  to  whom  it  is  said 
the  Irish  Parliament  awarded  a  pension  equal  to  the  inter- 
est on  twenty  thousand  pounds  as  a  recognition  of  the  great 
value  of  his  work. 

In  the  April,  1879,  number  of  Van  Nostrand's  Engineering 
Magazine  a  new  rule  was  proposed  by  Professor  J.  Wood- 
bridge  Davis,  which  is  less  laborious  than  the  old  rule. 
The  author  used  it  instead  of  Double  Meridian  Distances 
during  those  years  in  which  much  of  his  work  was  surveying 
and  cannot  understand  why  the  old  rule  is  not  abandoned. 
The  newer  one  is  called  the  Total  Latitude  rule  and  will 
alone  be  given  in  this  book.  It  is  based  on  trapezoids 
instead  of  triangles. 

Rule.  —  Multiply  the  total  latitude  of  each  station  by  the 
sum  of  the  departures  of  the  two  adjacent  courses.  The 
algebraic  half  sum  of  these  products  is  the  area. 

Algebraic  sum  must  first  be  understood  and  the  rules 
memorized. 

To  add  positive  (+)  and  negative  (  —  )  quantities,  subtract 
the  lesser  from  the  greater  and  prefix  to  the  result  the  sign  of 
the  greater.  This  has  been  illustrated  in  the  use  of  the 
traverse  table. 

To  subtract  positive  and  negative  quantities  change  the  sign 
of  the  subtrahend  and  then  proceed  as  in  addition. 

To  multiply  positive  and  negative  quantities  the  rules 
are: 

(a)  Like  signs  produce  plus. 

(+5)  X  (+6)  =  +30. 
(-5)  X  (-6)  =  +30. 

(b)  Unlike  signs  produce  minus. 

(-6)  =-30. 


In  practical  work  the  positive  (+)  sign  is  generally  under- 
stood so  is  seldom  written. 

The  following  example  is  taken  from  the  article  of 
Professor  Davis. 


132 


PRACTICAL  SURVEYING 


Sta. 

Bearing. 

Dist. 

Lat. 

Dep. 

Total 
lat. 

Adj. 
dep. 

Double 
areas. 

N+ 

2.21 
O.IS 

1-7* 

4.14 

S- 

E+ 

W- 

o 

i 

2 

3 
4 

N35°   E 
N83i°E 
S  57°   E 
S  34i°W 

Ns6*°w 

2.70 
1.29 

2.22 

3-55 

3-23 

I.2I 

2-93 

i-55 

1.28 
1.86 

4~69~ 

6.2543 
7.4104 
—  o.  1610 
8.3482 

2.OO 
2.69 
4.69 

Sq 

2.21 

2-36 

i.  «S 

-1.78 

iiare  cl: 

2.83 
3-14 
—  0.14 
-4.69 

.ains 

4.14 

2)21  .8519 
10.9259 

Area  1.0926  acres. 

When  using  the  Double  Meridian  Distance  rule  certain 
considerable  advantage  is  secured  by  starting  the  work  of 
computation  at  the  most  easterly  or  the  most  westerly 
station.  When  using  the  Total  Latitude  rule  any  station 
may  be  used  as  a  starting  point  but  if  that  station  is  chosen 
for  which  the  latitude  is  most  nearly  the  average  of  all,  the 
fewest  possible  figures  will  be  used  in  the  double  area 
factors. 

In  the  example  the  work  was  performed  as  follows: 

Total  latitude:  \2  2l 

+Q.I5- 
+2.36 

—  1.21 


-2.93 

-  1  .  78  Check 

Check.  —  The  total  latitude  for  the  last  station  must  be 
equal  to  the  latitude  of  that  station,  with  opposite  sign. 
The  total  latitude  of  the  first  station  is  zero. 

The  adjacent  departure  for  the  first  station  is  zero. 
2nd  station. 


Dep.  o 
Dep.  i 
Dep.  i 
Dep.  2 
Dep.  2 
Dep.  3 
Dep.  3 
Dep.  4 

=  +1-55 
=  +1.28 

+  2.83 
+  3-14 

—  o.  14 

-4.69 

=  +1.28 
=  +1.86 
=  +1.86 

=    —  2.OO 

=    —  2.OO 
=    —2.69 

COMPASS   SURVEYING  133 

The  only  checks  on  the  adjacent  departures  addition  are 
care  and  re-computing. 

Following  the  algebraic  law  for  dealing  with  positive  and 
negative  signs  the  column  of  double  areas  is  rilled.  For 
Sta.  i  and  2  the  sign  is  -f  for  both  the  total  latitude 
and  the  adjacent  departure,  therefore  the  double  area  is 
positive.  For  Sta.  3  the  first  is  +  and  the  second  is  — , 
therefore  the  double  area  is  negative.  For  Sta.  4  both 
signs  are  —  and  the  double  area  is  positive. 

DISTRIBUTING  ERRORS 

When  measuring  with  a  chain,  using  a  compass  for  the 
angles,  lengths  are  usually  taken  to  the  nearest  link. 
Some  surveyors  record  the  nearest  half  link  but  when  this 
'is  done  they  attempt  to  read  the  bearing  more  closely  than 
a  quarter  of  a  degree  by  estimating  the  position  of  the  north 
end  of  the  needle  between  graduations. 

The  maximum  error  in  angle  when  the  nearest  quarter 
of  a  degree  is  recorded  is  seven  minutes,  corresponding  to  a 
departure  of  0.2  link  per  chain  or  I  in  500.  The  maximum 
allowable  error  in  chaining  for  ordinary  farming  land  should 
not  exceed  I  in  500,  so  that  the  work  of  chainmen  having 
some  experience  is  about  equal  in  accuracy  to  that  of 
ordinary  compass  work. 

The  errors  here  referred  to  are  the  accidental  errors  of 
the  work.  Errors  due  to  a  chain  or  tape  being  longer  or 
shorter  than  standard  do  not  show  until  a  re-survey  is 
made  with  a  standard. 

Minor  errors  of  closure  are  distributed  proportionately 
to  lengths  in  compass  surveys,  the  effect  being  to  change 
bearings  and  distances.  The  transit  being  an  instrument 
by  means  of  which  angles  can  be  read  very  closely,  all 
closing  errors  are  distributed  over  the  lengths  only,  it  being 
easily  possible  to  check  the  angles  and  obtain  exact  angular 
closures. 

The  method  here  given  for  distributing  errors  is  applicable 
alike  to  compass  work  or  transit  work  since  it  is  bad 
practice  to  alter  bearings  and  distances  when  the  error  of 
closure  is  found  to  be  within  reasonable  limits.  Forcing  a 
closure  is  called  "fudging"  and  notes  of  a  compass  survey 


134 


PRACTICAL   SURVEYING 


are  looked  upon  with  suspicion  when  they  close  without 
error. 

Example.  —  A  field  was  surveyed  and  the  following  notes 
recorded : 


0 

N        c 

'    £.. 

Chains 
2.  71 

TV  8v 

t°  E 

I    "*O 

1 

S  S7' 

1    E" 

2   21 

7 

5  34^ 

\°  W  

•I     C/1 

4. 

TV& 

\cw.., 

.1.22 

When  the  latitudes  and  departures  were  computed  the 
field  did  not  close.  Before  the  area  of  a  field  can  be  com- 
puted the  northings  must  equal  the  southings  and  the 
eastings  must  equal  the  westings.  In  other  words,  in 
measuring  the  boundaries  of  a  field  the  surveyor  goes 
as  far  north  as  he  goes  south  and  as  far  east  as  he"1  goes 
west. 


Sta. 

Bearing. 

Dist. 

Lat. 

Dep. 

Balanced 

Lat. 

Dep. 

N+ 

S- 

E+ 

W- 

N+ 

S- 

E+ 

W- 

o 
i 

2 

3 

4 

N3S°  E 
N  83*°  E 
S  57°  E 
S34l°W 
NS6i0W 

2.73 
1.30 

2.21 

3.54 

3-22 

2.214 
0.147 

i  209 

i  -SSo 
1.292 
i  862 

2.2132 
0.1469 

1.2094 
2.9271 

I.S456 
1.2883 
1.8566 

1.9978 
2.6927 

1.777 

2.926 

1.992 
2.685 

1.7764 

13.00     4.138    4.135    4.704    4.677    4.1365  4.1365  4.6905  4.6905 
4.135  4.677 


Error  in  lat.=  0.003 


o .  027  =  Error  in  dep. 


/O  -»2     i     o  72 

Error  of  closure  =  V  -^— - — —  =  0.00208, 
V        1300 


— -  =  i  in  480. 
0.00208 

The  error,  being  within  reasonable  limits,  is  distributed 
as  follows: 


COMPASS   SURVEYING 


Latitudes 


'-       =  4.1365  correct  sum. 

4-1380 
4-I365 
0.0015  to  be  subtracted  from  the  northings. 

4-1365 
4-I350 
0.0015  to  be  added  to  the  southings. 

.  0.0015 

I  H  --  —  =  0.00036  to  increase  southings. 
4-I3e>5 

0.0015 

i  ---  —-  =  i  —  0.00036  =  0.99964  to  decrease  northings. 
4-I3t>5 


Northings. 

Southings. 

2.  214  X  O.  99964 

2.  2132 

o.  147  X  o.  99964 

o.  1469 

i  .  209  X  i  .  00036 

I  2019 

2.  926  X  I.  00036 

2.  9271 

4-  1365 

4-  1365 

The  results  are  placed  in  the  "Balanced"  columns. 
Departures 


—  --  =  4.6905  correct  sum. 

4.7040 
4-69Q5 
0.0135  to  be  subtracted  from  eastings. 

4-6905 

4-677 

0.0135  to  be  added  to  westings. 

i  +    '     ^  =  1.00287  to  increase  westings. 
1  ~  4^6905  =  a"713  to  decrease  eastings. 


136 


PRACTICAL  SURVEYING 


Eastings. 

Westings. 

i  .55    X  0.99713 

I  .  5456 

1.292  X  0.99713 

I  .  2883 

i  862  X  o  99713 

1.8566 

i  .992  X  i  .00287 
2.685  X  1-00287 

1-9978 
2.6927 

4-6905 

4-6905 

In  actual  practice  few  surveyors  do  all  the  figuring  here 
shown,  the  corrections  being  made  mentally  and  distrib- 
uted in  approximately  proportionate  amounts.  The  practi- 
cal surveyor  also  pays  great  attention  to  the  weighting  of 
the  various  courses  when  his  chainmen  are  inexperienced 
and  their  work  is  therefore  not  wholly  reliable. 

OMISSIONS 

It  is  not  possible  always  to  measure  each  side  of  a  field 
or  take  every  bearing  and  the  missing  parts  must  be  sup- 
plied by  platting  or  by  computation,  this  latter  method 
requiring  a  knowledge  of  trigonometry.  If  a  map  is  not 
accurately  made  to  a  very  large  scale  omissions  cannot 
be  obtained  from  it  within  the  limits  of  even  reasonable 
accuracy.  For  the  purpose,  however,  of  computing  areas 
by  mensuration  missing  lines  may  be  supplied  by  plotting. 
The  methods  to  be  now  described  should  be  carefully 
studied  and  worked  out  in  full  by  the  student,  for  he  will 
then  fully  understand  the  methods  of  computation  in  the 
chapter  following. 

There  are  six  cases  in  omissions 
and  one  field  will  be  used  to  illus- 
trate them. 

Chains 

0.  N  i6%°E 22.00 

1.  N82°   E T9.6o 

2.  S  17°   E 24.00 

3-  S  37°    W 22.00 

4.  #49°    W 25.20 


COMPASS   SURVEYING 


CASE  I.  —  Bearing  and  distance  omitted.     Fig.  141. 

Assume  the  course  from  2  to  3,  5  17°  E,  24  chains  to  be 
omitted. 

Plat  the  two  courses  o  and  I,  beginning  at  Sta.  o.  Re- 
verse the  bearings  for  courses  3  and  4  so  they  may  be 
platted  beginning  at  Sta.  o.  The  line  from  2  to  3  may 
then  be  drawn.  A  close  check  on  the  bearing  may  be  ob- 
tained by  drawing  a  meridian  line  through  one  end  of  the 
line  and  applying  a  protractor. 


FIG.  141. 


FIG.  142. 


CASE  II. — Bearing  read  but  length  not  measured.  Fig.  142. 

The  omitted  length  is  for  course  2.  Begin  at  Sta.  o  and 
lay  off  the  bearings  and  distances  to  Sta.  2  and  from  Sta.  2 
lay  off  by  the  protractor  a  long  line 
having  the  bearing  S  17°  E.  Re- 
versing the  bearings  of  courses  3  and 
4  as  in  the  first  example,  plot  them; 
Sta.  3  should  be  on  the  long  line 
drawn  from  Sta.  2  provided  the  field 
work  and  platting  are  correct. 

CASE   III.  —  Length   measured, 
bearing  not  read.     Fig.  143. 

Assume  same  course  to  be 
affected.  Beginning  at  Sta.  o  plot 
courses  o  and  I  in  a  forward  direc- 
tion and  courses  3  and  4  reversed. 
From  Sta.  2  as  a  center  with  a 
radius  equal  to  24  chains,  describe  an  arc.  If  the  work 
was  properly  done  the  arc  should  pass  through  Sta.  3. 


FIG.  143. 


138 


PRACTICAL  SURVEYING 


CASE  IV.  —  Two  bearings  read  but  lengths  not  measured. 

(a)  Adjacent  courses.     Fig.  144. 

The  omitted  lengths  are  assumed  to  be  for  courses  2  and 
3.  Begin  at  Sta.  o  and  plot  in  a  forward  direction  to  Sta.  2, 
from  which  point  draw  a  long  line  on  the  given  bearing. 
Then  plot  the  reversed  courses;  the  intersection  of  the 
long  line  from  Sta.  2  with  that  from  Sta.  4  fixes  the  loca- 
tion of  Sta.  3. 

(b)  Courses  not  adjacent.     Fig.  145. 

Assume  the  omitted  lengths  to  be  in  courses  I  and  4. 
Begin  at  o  and  lay  off  the  line  from  o  to  I .  The  length  of 


the  next  line  is  not  known  so  this  course  is  disregarded 
and  from  Sta.  I  lay  off  the  bearing  and  distance  of  course 
2.  5"  17°  E  24  chains  and  follow  that  with  the  next  course. 
Then  from  o  lay  off  the  bearing  5  82°  W  (the  reverse  of 
N  82°  E).  From  4'  lay  off  the  bearing  N  49°  W.  These 
lines  will  intersect  at  i'.  By  using  triangles  transfer  the 
line  I '-2'  to  1-2  and  thus  locate  2.  From  2  draw  2-3  equal 
and  parallel  to  1-3'  and  from  3  draw  line  3-4  equal  and 
parallel  to  3'~4'.  Then  the  line  4-0  will  be  equal  and 
parallel  to  Iine4'-i/. 

CASE  V.  —  Two  lengths  measured  but  bearings  not 
read. 

(a)  Adjacent  courses.     Fig.  146. 

Assuming  the  bearings  of  courses  2  and  3  to  have  been 
omitted  plot  the  other  courses  from  Sta.  o  as  before.  From 


COMPASS   SURVEYING 


139 


Sta.  2  describe  an  arc  with  a  radius  equal  to  the  length  of 

course  2  and  from  Sta.  4  describe  an  arc  with  a  radius 

equal    to    the    length    of   course   3. 

These  arcs  will  intersect  in  3  and  in 

3'.     It  will  be  necessary  to  view  the 

land  in  order  to  know  which  point 

is  correct. 

(b)  Courses  not  adjacent.  Fig.  147. 

The  bearings  of  courses  2  and  4 
were  not  read.  The  remaining  three 
courses  are  plotted  from  o  to  I,  I 
to  2,  2  to  3'.  From  o  an  arc  with 
radius  equal  to  the  length  of  course 
4  is  described  and  from  3'  an  arc  with 
radius  equal  to  the  length  of  course 
2  is  described.  Fig.  147  (a)  shows 
the  work,  (b)  the  figure  resulting  when  courses  1-2  and  0—4 
are  omitted  and  (c)  the  figure  resulting  when  o-i  and  4-3 
are  omitted.  In  this  case  a  knowledge  of  the  shape  of  the 
field  is  necessary  in  order  to  know  which  solution  is  right. 


FIG.  146. 


co; 


CASE  VI.  —  Bearing  of  one  course  read  but  length  not 
measured  and  length  measured  but  bearing  omitted  on 
another  course. 

(a)  Adjacent  courses.     Fig.  148. 

Plat  as  before.  The  bearing  from  Sta.  2  having  been 
read  lay  off  a  dotted  line.  The  length  of  course  3  having 
been  measured  describe  an  arc  with  this  radius  from  Sta.  4. 
This  gives  two  possible  solutions  so  that  actual  knowledge 
of  the  shape  of  the  field  is  necessary. 


140 


PRACTICAL   SURVEYING 


(b)  Courses  not  adjacent.     Fig.  149. 

It  is  assumed  that  the  bearing  from  Sta.  I  was  omitted 
and  the  length  of  course  4  was  omitted. 

Plot  the  course  o-i.  From  Sta.  I  describe  an  arc  with 
radius  equal  to  the  length  of  course  I  and  from  o  lay  off 
the  line  0-4  with  the  bearing  of  course  4.  On  a  piece  of 


FIG.  148. 


tracing  paper  plot  the  line  3-4  and  from  3  the  line  3-2, 
extending  the  line  considerably  beyond  the  mark  at  2. 
From  4  plot  a  line  having  the  bearing  of  course  4.  Place 
the  line  4-0  on  the  tracing  paper  over  the  same  line  on  the 
drawing  paper  and  move  it  along  this  line  until  the  line 
2—3  is  tangent  to  the  arc  and  the  point  2  is  on  the  arc.  A 
needle  put  through  at  2,  3  and  4  will  mark  the  stations  and 
lines  may  then  be  drawn  to  complete  the  map. 

A  careful  examination  of  the  problems  given  shows  that 
in  reality  there  are  only  four  "cases,  in  which  the  lost,  or 
omitted,  parts  may  be: 

CASE  I.  —  Bearing  and  length  of  one  course. 

CASE  II.  —  Lengths  of  two  courses. 

CASE  III.  —  Bearing  of  two  courses. 

CASE  IV.  —  Length  of  one  course  and  bearing  of  another. 


COMPASS   SURVEYING 
PROBLEMS 


141 


Find  by  computation  the  areas  of  the  following  fields. 
Plat  same  carefully  and  check  the  computations  by  men- 
suration. 

Chains 

1.  o.    N  i6£°  E 22.00 

1.  NS2°    E 19-60 

2.  5  17°    E 24.00 

3-537°      W 22.00 

4.    N 49°    W 25.20 

Chains 

2.  o.   #15°    E 80 

1.  #37!°  E«* 40 

2.  East 30 

3.  S  11°    E 50 

4.  South 54 

5.  West 40 

6.  S  36i°  W 40 

7-  N&VW 34 

Chains 

3.  o.   #75°    E 13.70 

1.  N  20\°  E 10.30 

2.  East 16.20 

3-    S  33J"  W 35-30 

4.576°    W 16.00 

5.   North 9.00 

6.584°    W ii.  60 

7-    Ns&W ii. 60 

8.  N  36! °  E 19.  20 

9.  N  22\°  E 14.00 

10.  5  76!°  E 12.00 

11.  5  15°    W 10.85 

12.  5  i6f  °  W 10.12 


142 


PRACTICAL  SURVEYING 


TRAVERSE  TABLE 


Course. 

Dist.  i. 

Dist.  2. 

Dist.  3. 

Dist.  4. 

Dist.  5. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

o  IS 

I.OOOO 

0.0044 

2.OOOO 

0.0087 

3.0000 

0.0131 

4.0000 

0.0175 

5.0000 

0.0218 

89  45 

30 

0000 

0087 

1.9999 

0175 

2.9999 

0262 

3-9998 

0349 

4.9998 

0436 

30 

45 

0.9999 

0131 

9998 

0262 

9997 

0393 

9997 

0524 

9996 

0654 

15 

I   0 

9998 

0175 

9997 

0349 

9995 

0524 

9994 

0698 

9992 

0873 

89  o 

IS 

9998 

0218 

9995 

0436 

9993 

0654 

9990 

0873 

9988 

1091 

45 

30 

9997 

0262 

9993 

0524 

9990 

0785 

9986 

1047 

9983 

1309 

30 

45 

9995 

0305 

9991 

0611 

9986 

0916 

9981 

1222 

9977 

1527 

15 

2   0 

9994 

0349 

9988 

0698 

9982 

1047 

9976 

1396 

9970 

1745 

88  o 

IS 

9992 

0393 

9985 

0785 

9977 

1178 

9969 

1570 

996i 

1963 

45 

30 

9990 

0436 

9981 

0872 

9971 

1309 

9962 

1745 

9952 

2181 

3O 

45 

0.9988 

0.0480 

1.9977 

0.0960 

2.9965 

0.1439 

3-9954 

0.1919 

4-9942 

0.2399 

15 

3   o 

9986 

0523 

9973 

1047 

9959 

1570 

9945 

2093 

9931 

2617 

87  o 

15 

9984 

0567 

9968 

H34 

9952 

1701 

9936 

2268 

9920 

2835 

45 

30 

9981 

0610 

9963 

1221 

9944 

1831 

9925 

2442 

9907 

3052 

30 

45 

9979 

0654 

9957 

1308 

9936 

1962 

9914 

2616 

9893 

3270 

15 

4   o 

9976 

0698 

9951 

1395 

9927 

2093 

9903 

2790 

9878 

3488 

86  o 

IS 

0741 

9945 

1482 

99i8 

2223 

9890 

2964 

9863 

3705 

45 

30 

9969 

0785 

9938 

1569 

9908 

2354 

9877 

3138 

9846 

3923 

30 

45 

9966 

0828 

9931 

1656 

9897 

2484 

9863 

3312 

9828 

4140 

15 

5   o 

9962 

0872 

9924 

1743 

9886 

2615 

9848 

3486 

9810 

4358 

85  o 

IS 

0.9958 

0.0915 

1.9916 

0.1830 

2.9874 

0.2745 

3.9832 

0.3660 

4.9790 

0.4575 

45 

30 

9954 

0958 

1917 

9862 

2875 

9816 

3834 

9770 

4792 

30 

45 

9950 

1  002 

9899 

2004 

9849 

3006 

9799 

4008 

9748 

15 

6  o 

9945 

1045 

9890 

2091 

9836 

3136 

978i 

4181 

9726 

5226 

84  o 

IS 

9941 

1089 

9881 

2177 

9822 

3266 

9762 

4355 

9703 

5443 

45 

30 

9936 

1132 

9871 

2264 

9807 

3390 

9743 

4528 

9679 

5660 

3o 

45 

9931 

1175 

9861 

2351 

9792 

9723 

4701 

9653 

5877 

15 

7   o 

9925 

1219 

9851 

2437 

9776 

3656 

9702 

4875 

9627 

6093 

83  o 

IS 

9920 

1262 

9840 

2524 

9760 

3786 

9680 

5048 

9600 

6310 

45 

30 

9914 

1305 

9829 

2611 

9743 

39i6 

9658 

5221 

9572 

6526 

30 

45 

0.9909 

0.1349 

1.9817 

0.2697 

2.9726 

0.4046 

3.9635 

0.5394 

4-9543 

0.6743 

IS 

8   o 

9903 

1392 

9805 

2783 

9708 

4175 

9611 

5567 

9513 

6959 

82  o 

IS 

9897 

1435 

9793 

2870 

9690 

4305 

9586 

5740 

9483 

7175 

45 

30 

1478 

978o 

2956 

9670 

4434 

956i 

5912 

9451 

7390 

30 

45 

9884 

1521 

9767 

3042 

9651 

4564 

9534 

6085 

9418 

7606 

IS 

9  o 

9877 

1564 

9754 

3129 

9631 

4693 

9508 

6257 

9384 

7822 

81  o 

IS 

9870 

1607 

9740 

3215 

9610 

4822 

9480 

6430 

9350 

8037 

45 

30 

9863 

1650 

9726 

3301 

9589 

4951 

9451 

6602 

9314 

8252 

30 

45 

9856 

1693 

971  1 

3387 

9567 

5080 

9422 

6774 

9278 

8467 

15 

10  o 

9848 

1736 

9696 

3473 

9544 

5209 

9392 

6946 

9240 

8682 

80  o 

IS 

0.9840 

0.1779 

1.9681 

0.3559 

2.9521 

o  .  5338 

3.9362 

0.7118 

4.9202 

0.8897 

45 

30 

9833 

1822 

9665 

3645 

9498 

5467 

9330 

7289 

9163 

9112 

30 

45 

9825 

1865 

9649 

3730 

9474 

5596 

9298 

7461 

9123 

9326 

IS 

II   0 

9816 

1908 

9633 

3816 

9449 

5724 

9265 

7632 

9081 

9540 

79  o 

IS 

9808 

1951 

9616 

3902 

9424 

5853 

9231 

7804 

9°39 

9755 

45 

30 
45 

9799 
9790 

1994 
2036 

9598 
958i 

3987 

4073 

9398 
9371 

598i 
6109 

9197 
9162 

7975 
8146 

8952 

9968 
1.0182 

30 
IS 

12  O 

978i 

2079 

9563 

4158 

9344 

6237 

9126 

8316 

8907 

0396 

78  o 

•  is 

9772 

2122 

9545 

4244 

9317 

6365 

9089 

848? 

8862 

0609 

45 

30 

9763 

2l64 

9526 

4329 

9289 

6493 

9052 

8658 

8815 

0822 

30 

45 

0.9753 

0.2207 

1.9507 

0.4414 

2.9260 

0.6621 

3.9014 

0.8828 

4.8767 

I  •  1035 

IS 

13  0 

9744 

2250 

9487 

4499 

9231 

6749 

8975 

8998 

8719 

1248 

77  o 

IS 

9734 

2292 

9468 

4584 

9201 

6876 

8935 

9168 

8669 

1460 

45 

30 

9724 

2334 

9447 

4669 

9171 

7003 

8895 

9338 

8618 

1672 

30 

45 

9713 

2377 

9427 

4754 

9140 

7131 

8854 

9507 

8567 

1884 

15 

14  0 

9703 

2419 

9406 

4838 

9109 

7258 

8812 

9677 

8515 

2096 

76  o 

15 

2462 

9385 

4923 

9077 

7385 

8769 

9846 

8462 

2308 

45 

30 

9681 

2504 

9363 

5008 

9044 

75" 

8726 

1.0015 

8407 

2519 

30 

45 

9670 

2546 

9341 

5092 

9011 

7638 

8682 

0184 

8352 

2730 

IS 

15  o 

9659 

2588 

9319 

5176 

8978 

7765 

8637 

0353 

8296 

2941 

75  o 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep.  Lat. 

Dep. 

Lat. 

Course. 

Dist.  I. 

Dist.  2. 

Dist.  3. 

Dist.  4. 

Dist.  5. 

COMPASS   SURVEYING 


TRAVERSE  TABLE  (Continued) 


Dist.  6. 

Dist.  7. 

Dist.  8. 

Dist.  9- 

Dist.  10. 

Course 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

o  IS 

5-9999 

0.0262 

6.9999 

0.0305 

7-9999 

0.0349 

8.9999 

0.0393 

9-9999 

0.0436 

89  45 

30 

9998 

0524 

9997 

0611 

9997 

0698 

9997 

0785 

9996 

0873 

30 

45 

9995 

0785 

9994 

0916 

9993 

1047 

9992 

1178 

9991 

1309 

IS 

I    0 

9991 

1047! 

9989 

1222 

9988 

1396 

9986 

IS7I 

9985 

1745 

89  o 

IS 
30 

9986 
9979 

1309 
1571  1 

1527 
1832 

998i 
9973 

1745 

2094 

9979 
9969 

1963 
2356 

9966 

2181 
2618 

45 
30 

45 

9972 

1832 

9967 

2138 

9963 

2443 

9958 

2748 

9953 

3054 

IS 

2   0 

9963 

2094 

9957 

2443 

9951 

2792 

9945 

3i4i 

9939 

3490 

88  o 

IS 

9954 

2356 

9946 

2748 

9938 

3141 

9931 

3533 

9923 

3926 

45 

30 

9943 

2617! 

9933 

3053 

9924 

3490 

9914 

3926| 

9905 

4362 

30 

45 

5-9931 

0.2879 

6.9919 

0.3358 

7.9908 

0.3838 

8.9896 

0.4318 

9  9885 

0.4798 

15 

3  0 

9918 

3140 

9904 

3664 

9890 

4187 

9877 

4710 

'9863 

5234 

87  o 

IS 

9904 

3402 

9887 

3968 

9871 

4535 

9855 

5102 

9839 

5669 

45 

30 

9888 

3663 

9869 

4273 

9851 

4884 

9832 

5494 

98i3 

6105 

30 

45 

9872 

3924 

9850 

4578 

9829 

5232 

9807 

5836 

9786 

6540 

IS 

4  o 

9854 

4185 

9829 

4883 

9805 

5S8i 

978i 

6278 

9756 

6976 

86  o 

IS 

9835 

4447 

9808 

5188 

978o 

5929 

9753 

6670 

9725 

7411 

45 

30 

98i5 

4708 

9784 

5492 

9753 

6277 

9723 

7061 

9692 

7846 

30 

45 

9794 

4968 

9760 

5797 

9725 

6625 

9691 

7453 

9657 

8281 

IS 

5  0 

9772 

5229' 

9734 

6101 

9696 

6972 

9658 

7844 

96i9 

8716 

85  o 

IS 

5.9748 

0.5490' 

6.9706 

0.6405 

7.9664 

0.7320 

8.9622 

0.8235 

9.958o 

0.9150 

45 

30 

9724 

5751 

9678 

6709 

9632 

7668 

9586 

8626 

9540 

9585 

30 

45 

9698 

6011 

9648 

7013 

9597 

8015 

9547 

9017 

9497 

1.0019 

IS 

6  o 

9671 

6272 

96i7 

7317 

9562 

8362 

9507 

9408 

9452 

0453 

84  o 

IS 

9643 

6532 

9584 

7621 

9525 

8709 

9465 

9798 

9406 

'0887 

45 

30 

9614 

6792 

9550 

7924 

9486 

9056 

9421 

1.0188 

9357 

1320 

30 

45 

9584 

7052 

9515 

8228 

9445 

9403 

9376 

0578 

9307 

1754 

IS 

7  o 

9553 

7312 

9478 

8531 

9404 

9750 

9329 

0968 

9255 

2187 

83  o 

IS 

9520 

7572 

9440 

8834 

936o 

1.0096 

9280 

1358 

9200 

2620 

45 

30 

9487 

7832 

9401 

9137 

9316 

0442 

9230 

1747 

9144 

3053 

30 

45 

5-9452 

0.8091 

6.9361 

0.9440 

7.9269 

1.0788 

8.9178 

I.  2137 

9-9087 

1.3485 

15 

8  o 

9416 

8350 

9319 

9742 

9221 

"34 

9124 

2526 

9027 

3917 

82  o 

IS 

9379 

8610 

9276 

1.0044 

9172 

1479 

9069 

2914 

8965 

4349 

45 

3° 

9341 

8869 

9231 

0347 

9121 

1825 

9011 

3303 

8902 

478i 

30 

45 

9302 

9127 

9185 

0649 

9069 

2170 

8953 

3691 

8836 

5212 

IS 

9  o 

9261 

9386 

9138 

0950 

9015 

2515 

8892 

4079 

8769 

5643 

8l  o 

15 

9220 

9645 

9090 

1252 

8960 

2859 

8830 

4467 

8700 

6074 

45 

30 

9177 

9903 

9040 

1553 

8903 

3204 

8766 

4854 

8629 

6505 

30 

45 

9133 

1.0161 

8989 

1854 

8844 

3548 

8700 

5241 

8556 

6935 

IS 

10  o 

9088 

0419 

8937 

2155 

8785 

3892 

8633 

5628 

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7365 

80  0 

is 

5.9042 

1.0677 

6.8883 

1.2456 

7-8723 

1.4235 

8.8564 

1.6015 

9.8404 

1-7794 

45 

30 

8995 

0943 

8828 

2756 

8660 

4579 

8493 

6401 

8325 

8224 

30 

45 

8947 

1191 

8772 

3057 

8596 

4922 

8421 

6787 

8245 

8652 

IS 

II   0 

8898 

1449 

8714 

3357 

8530 

5265 

8346 

7173 

8163 

9081 

79  o 

IS 

8847 

1705 

8655 

3656 

8463 

5607 

8271 

7558 

8079 

9509 

45 

30 

8795 

1962 

8595 

3956 

8394 

5949 

8193 

7943 

7992 

9937 

30 

45 

8743 

2219 

8533 

4255 

8324 

6291 

8114 

8328 

7905 

2.0364 

IS 

12  O 

8689 

2475 

8470 

4554 

8252 

6633 

8033 

8712 

7815 

0791 

78  o 

IS 

8634 

2731 

8406 

4852 

8178 

6974 

7951 

9096 

7723 

1218 

45 

30 

8578 

2986 

8341 

5151 

810^ 

7315 

7867 

9480 

7630 

1644 

30 

45 

5.8521 

1.3242 

6.8274 

1.5449 

7.8027 

1.7656 

8.7781 

1.9863 

9-7534 

2  .  2O70 

IS 

13  o 

8462 

3497 

8206 

5747 

7950 

7996 

7693 

2.0246 

7437 

2495 

77  o 

IS 

8403 

3752 

8i37 

6044 

7870 

8336 

7604 

0628 

7338 

2920 

45 

30 

8342 

4007 

8066 

6341 

7790 

8676 

7513 

IOIO 

7237 

3345 

30 

45 

8281 

4261 

7994 

6638 

7707 

9015 

7421 

1392 

7134 

3769 

IS 

14  o 

8218 

4515 

7921 

6935 

7624 

9354 

7327 

1773 

7030 

4192 

76  o 

IS 

8154 

4769 

7846 

7231 

7538 

9692 

7231 

2154 

6923 

4615 

45 

30 

8089 

5023 

7770 

7527 

7452 

2.0030 

7133 

2534 

6815 

5038 

30 

45 

8023 

5276 

7693 

7822 

7364 

0368 

7034 

2914 

6705 

5460 

15 

IS  o 

7956!  5529 

7615 

8117 

7274 

0706 

6933 

3294 

6593 

5882 

75  o 

Dep.  Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dist.  6. 

Dist.  7. 

Dist.  8. 

Dist.  9. 

Dist.  ic. 

Course. 

144 


PRACTICAL  SURVEYING 


TRAVERSE  TABLE  (Continued) 


Course. 

Dist.  I. 

Dist.  2. 

Dist.  3. 

Dist.  4. 

Dist.  5. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

IS  IS 

0.9648 

0.2630 

1.9296 

0.5261 

2.8944 

0.7891 

3.8591 

1.0521 

4-8239 

1.3152 

74  45 

30 

9636 

2672 

9273 

5345 

8909 

8017 

8545 

0690 

8182 

3362 

30 

45 

9625 

2714 

9249 

5429 

8874 

8i43 

8498 

0858 

8123 

3572 

15 

16  o 

96i3 

2756 

9225 

5513 

8838 

8269 

8450 

1025 

8063 

3782 

74  o 

IS 

9600 

2798 

9201 

5597 

8801 

8395 

8402 

H93 

8002 

3991 

45 

30 

9588 

2840 

9176 

5680 

8765 

8520 

8353 

1361 

794i 

4201 

30 

45 

9576 

2882 

9i5i 

5764 

8727 

8646 

8303 

1528 

7879 

4410 

15 

17  o 

9563 

2924 

9126 

5847 

8689 

877i 

8252 

1695 

7815 

4619 

73  o 

IS 

9550 

2965 

9100 

5931 

8651 

8896 

8201 

1862 

7751 

4827 

45 

30 

9537 

3007 

9074 

6014 

8612 

9021 

8149 

2028 

7686 

5035 

30 

45 

0.9524 

0.3049 

1.9048 

0.6097 

2.8572 

0.9146 

3.8096 

1.2195 

4.7620 

I.S243 

IS 

18  o 

9Si  i 

3090 

9021 

6180 

8532 

9271 

8042 

2361 

7553 

5451 

72  0 

IS 

9497 

3132 

8994 

6263 

8491 

9395 

7988 

2527 

7485 

5658 

45 

30 

9483 

3173 

8966 

6346 

8450 

9519 

7933 

2692 

74i6 

586s 

30 

45 

9469 

3214 

8939 

6429 

8408 

9643 

7877 

2858 

7347 

6072 

15 

19  o 

9455 

3256 

8910 

6511 

8366 

9767 

7821 

3023 

7276 

6278 

71  o 

IS 

9441 

3297 

8882 

6594 

8323 

9891 

7764 

3188 

7204 

6485 

45 

30 

9426 

3338 

8853 

6676 

8279 

1.0014 

7706 

3352 

7132 

6690 

30 

45 

94" 

3379 

8824 

6758 

8235 

0138 

7647 

3517 

7059 

6896 

IS 

20  0 
IS 

9397 
0.9382 

3420 
0.3461 

8794 
1.8764 

6840 
0.6922 

8191 
2.8146 

0261 

7588 
3.7528 

3681 
1.3845 

6985 
4.6910 

7101 
1.7306 

70  o 
45 

30 

9367 

3502 

8733 

7004 

8100 

0506 

7467 

4008 

6834 

75io 

30 

45 

9351 

3543 

8703 

7086 

8054 

0629 

7405 

4172 

6757 

7715 

IS 

21  0 

9336 

3584 

8672 

7167 

8007 

0751 

7343 

4335 

6679 

7918 

69  o 

15 

9320 

3624 

8640 

7249 

796o 

0873 

7280 

4498 

6600 

8122 

45 

30 

9304 

3665 

8608 

7330 

7913 

0995 

7217 

4660 

6521 

8325 

30 

45 

9288 

3706 

8576 

74" 

7864 

1117 

7152 

4822 

6440 

8528 

IS 

22  0 

9272 

3746 

8544 

7492 

7816 

1238 

7087 

4984 

6359 

8730 

68  o 

IS 

9255 

3786 

8511 

7573 

7766 

1359 

7022 

5146 

6277 

8932 

45 

30 
45 

9239 
0.9222 

3827 
0.3867 

8478 
1.8444 

7654 
0.7734 

77i6 
2.7666 

1481 
1.1601 

5307 
1.5468 

6194 
4.6110 

9134 
1.9336 

30 
15 

23  o 

9205 

3907 

8410 

7815 

7615 

1722 

6820 

5629 

6025 

9537 

67  o 

IS 

9188 

3947 

8376 

7895 

7564 

1842 

6752 

5790 

5940 

9737 

45 

30 

9171 

3987 

8341 

7975 

7512 

1962 

6682 

5950 

5853 

9937 

30 

45 

9153 

4027 

8306 

8055 

7459 

2082 

6612 

6110 

5766 

2.0137 

IS 

24  o 

9135 

4067 

8271 

8i35 

7406 

22O2 

6542 

6269 

5677 

0337 

66  o 

IS 

9118 

4107 

8235 

8214 

7353 

2322 

6470 

6429 

5588 

0536 

45 

30 

9100 

4147 

8i99 

8294 

7299 

2441 

6398 

6588 

5498 

0735 

30 

45 

9081 

4187 

8163 

8373 

7244 

2560 

6326 

6746 

5407 

0933 

IS 

25  0 

9063 

4226 

8126 

8452 

7i89 

2679 

6252 

6905 

5315 

1131 

65  o 

IS 

0.9045 

0.4266 

1.8089 

0.8531 

2.7134 

1.2797 

3.6178 

1.7063 

4-5223 

2.1328 

45 

30 

9026 

4305 

8052 

8610 

7078 

2915 

6103 

7220 

5129 

1526 

30 

45 

9007 

4344 

8014 

8689 

7021 

3033 

6028 

7378 

5035 

1722 

IS 

26  o 

8988 

4384 

7976 

8767 

6964 

3151 

5952 

7535 

4940 

1919 

64  o 

15 

8969 

4423 

7937 

8846 

6906 

3269 

5875 

7692 

4844 

2114 

45 

30 

8949 

4462 

7899 

8924 

6848 

3386 

5797 

7848 

4747 

2310 

30 

45 

8930 

4501 

7860 

9002 

6789 

35O3 

5719 

8004 

4649 

2505 

IS 

27  o 

8910 

4540 

7820 

9080 

6730 

3620 

5640 

8160 

4550 

2700 

63  o 

15 

8890 

4579 

7780 

9157 

6671 

3736 

556i 

8315 

4451 

2894 

45 

30 

8870 

4617 

7740 

9235 

6610 

3852 

548o 

8470 

4351 

3087 

30 

45 

0.8850 

0.4656 

1.7700 

0.9312 

2.6550 

1.3968 

3-5400 

1.8625 

4.4249 

2.3281 

IS 

28  0 

8829 

4695 

7659 

9389 

6488 

4084 

53i8 

8779 

4147 

3474 

62  o 

IS 

8809 

4733 

7618 

9466 

6427 

4200 

5236 

8933 

4045 

3666 

45 

30 
45 

8788 
8767 

4772 
4810 

7576 
7535 

9543 
9620 

6365 
6302 

4315 
4430 

5069 

9086 
9240 

3941 
3836 

3858 
4049 

30 
IS 

29  o 

8746 

4848 

7492 

9696 

6239 

4544 

4985 

9392 

3731 

4240 

61  o 

IS 

8725 

4886 

7450 

9772 

6i75 

4659 

4900 

9545 

3625 

4431 

45 

30 

8704 

4924 

7407 

9848 

6m 

4773 

4814 

9697 

35i8 

4621 

30 

45 

8682 

4962 

7364 

9924 

6046 

4886 

4728 

9849 

34io 

4811 

IS 

30  o 

8660 

5000 

7321 

I.OOOO 

598T 

5000 

4641 

2.OOOO 

33QI 

SOOQ 

60  o 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Bourse" 

Dist.  i. 

Dist.  2. 

Dist.  3.      Dist.  4. 

Dist.  5. 

COMPASS   SURVEYING 


145 


TRAVERSE  TABLE  (Continued) 


Course. 

Dist.  6. 

Dist.  7. 

Dist.  8. 

Dist.  9. 

Dist.  10. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

0    / 

o   / 

IS  15 

S.7887 

1.5782 

6.7535 

i  .8412 

7.7183 

2  .  IO42 

8.6831 

2.3673 

9.6479 

2.6303 

74  45 

30 

7818 

6034 

7454 

8707 

7090 

1379 

6727 

4051 

6363 

6724 

30 

45 

7747 

6286 

7372 

9001 

6996 

1715 

6621 

4430 

6246 

7144 

IS 

16  o 

7676 

6538 

7288 

9295 

6901 

2051 

6514 

4807 

6126 

7564 

74  o 

15 

7603 

6790 

7203 

9588 

6804 

2386 

6404 

5185 

6005 

7983 

45 

30 

7529 

7041 

7117 

9881 

6706 

2721 

6294 

556i 

5882 

8402 

30 

45 

7454 

7292 

7030 

2.0174 

6606 

3056 

6181 

5938 

5757 

8820 

15 

17  o 

7378 

7542 

6941 

0466 

6504 

3390 

6067 

6313 

5630 

9237 

73  o 

IS 

7301 

7792 

6851 

0758 

6402 

3723 

5952 

6689 

5502 

9654 

45 

30 

7223 

8042 

6760 

1049 

6297 

4056 

5835 

7064 

5372 

3.0071 

30 

45 

5-7144 

1.8292 

6.6668 

2.1341 

7.6192 

2.4389 

8.5716 

2.7438 

9.5240 

3.0486 

IS 

18  o 

7063 

8541 

6574 

1631 

6085 

4721 

5S9S 

7812 

5106 

0902 

72  0 

IS 

8790 

6479 

1921 

5976 

5053 

5473 

8185 

4970 

1316 

45 

30 

6899 

9038 

6383 

221  1 

5866 

5384 

5349 

8557 

4832 

1730 

30 

45 

6816 

9286 

6285 

25OI 

5754 

5715 

5224 

8930 

4693 

2144 

IS 

19  o 

6731 

9534 

6186 

2790 

5641 

6045 

5097 

9301 

4552 

2557 

71  0 

IS 

6645 

978i 

6086 

3078 

5527 

6375 

4968 

9672 

4409 

2969 

45 

30 

6558 

2.0028 

5985 

5411 

6705 

4838 

3  0043 

4264 

338i 

30 

45 

6471 

0275 

5882 

3654 

5294 

7033 

4706 

0413 

4118 

3792 

15 

20  0 

6382 

0521 

5778 

3941 

5175 

7362 

4572 

0782 

3969 

4202 

70  0 

IS 

5.6291 

2.0767 

6.5673 

2.4228 

7-5055 

2.7689 

8.4437 

3-iiSi 

9.3819 

3.4612 

45 

'30 

6200 

1012 

5567 

4515 

-8017 

4300 

1519 

3667 

5021 

30 

45 

6108 

1257 

5459 

4800 

4811 

8343 

4162 

1886 

3514 

5429 

IS 

21  0 

6015 

1502 

5351 

5086 

4686 

8669 

4022 

2253 

3358 

5837 

69  o 

15 

5920 

1746 

5241 

5371 

456i 

8995 

3881 

2619 

3201 

6244 

45 

30 

5825 

1990 

5129 

5655 

4433 

9320 

3738 

2985 

3042 

6650 

30 

45 

5729 

2233 

5017 

4305 

9645 

3593 

33SO 

2881 

7056 

IS 

22  0 

5631 

2476 

4903 

6222 

4175 

9969 

3447 

3715 

2718 

746i 

68  o 

IS 

5532 

2719 

4788 

6505 

4043 

3.0292 

3299 

4078 

2554 

7865 

45 

30 

5433 

2961 

4672 

6788 

3910 

0615 

3149 

4442 

2388 

8268 

30 

45 

5-5332 

2.3203 

6.4554 

2.7070 

7.3776 

3-0937 

3-4804 

9  .  2220 

3.8671 

IS 

23  o 
IS 

5230 
SI27 

3444 
3685 

4435 
4315 

39 

3640 
3503 

1258 
1580 

2691 

Si66 
5527 

2O50 
1879 

9073 
9474 

67  o 
45 

30 

5024 

3925 

4194 

79" 

3365 

1900 

2535 

S887 

1706 

9875 

30 

45 

4919 

4165 

4072 

8192 

3225 

222O 

2378 

6247 

1531 

4.0275 

IS 

24  o 

4813 

4404 

3948 

8472 

3084 

2539 

2219 

6606 

1355 

0674 

66  o 

IS 

4706 

4643 

3823 

8750 

2941 

2858 

2059 

6965 

1176 

1072 

45 

30 

4598 

4882 

3697 

9029 

2797 

3175 

1897 

7322 

0996 

1469 

30 

45 

4489 

5120 

3570 

93o6 

2651 

3493 

1733 

7679 

0814 

1866 

IS 

25  o 

4378 

5357 

3442 

9583 

2505 

3809 

1568 

8036 

0631 

2202 

65  o 

IS 

5  .  4267 

2.5594 

6.3312 

2.9860 

7.2356 

3.4125 

8.1401 

3.8391 

9.0446 

4-2657 

45 

30 

4IS5 

5831 

3181 

3-0136 

4441 

1233 

8746 

0259 

3051 

30 

'  45 

4042 

6067 

3049 

0411 

2056 

4756 

1063 

9100 

0070 

3445 

IS 

26  o 

3928 

6302 

2916 

0686 

1904 

5070 

0891 

9453 

8.9879 

3837 

64  o 

IS 

3812 

6537 

2781 

0960 

1750 

5383 

0719 

9806 

9687 

4229 

45 

30 

3696 

6772 

2645 

1234 

1595 

5696 

0544 

4-0158 

9493 

4620 

30 

45 

3579 

7006 

2509 

1507 

1438 

6008 

0368 

0509 

9298 

5010 

IS 

27  o 

3460 

7239 

2370 

1779 

1281 

6319 

0191 

o8sQ 

9101 

5399 

63  o 

IS 

3341 

7472 

2231 

2051 

II2I 

6630 

0012 

1209 

8902 

5787 

45 

30 

3221 

7705 

2091 

2322 

0961 

6940 

7.9831 

1557 

8701 

6175 

30 

45 

5.3099 

2.7937 

6-1949 

3-2593 

7-0799 

3.7249 

7.9649 

4.1905 

8.8499 

4.6561 

IS 

28  o 

2977 

8168 

1806 

0636 

7558 

9465 

2252 

8295 

6947 

62  o 

IS 

2853 

8399 

1662 

3132 

0471 

7866 

9280 

2599 

8089 

7332 

45 

30 

2729 

8630 

1517 

3401 

0305 

8173 

9094 

2944 

7882 

7716 

30 

45 

2604 

8859 

1371 

3669 

0138 

8479 

8905 

3289 

7673 

8099 

IS 

29  o 

2477 

9089 

1223 

3937 

6.9970 

8785 

8716 

3633 

7462 

61  o 

15 

2350 

9317 

1075 

4203 

9800 

9090 

8525 

3976 

7250 

8862 

45 

30 

2221 

9545 

0925 

4470 

9628 

9394 

8332 

4318 

7036 

9242 

30 

45 

2092 

9773 

0774 

4735 

9456 

9697 

8138 

4659 

6820 

9622 

15 

30  0 

1962 

3.0000 

0622 

5000 

9282 

4.0000 

7942 

5000 

6603 

5.0000 

60  o 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dist.  6. 

Dist.  7. 

Dist.  8. 

Dist.  9. 

Dist.  10. 

L-oursc* 

I46 


PRACTICAL  SURVEYING 


TRAVERSE  TABLE  (Continued) 


Course. 

Dist.  I. 

Dist.  2. 

Dist.  3. 

Dist.  4. 

Dist.  5. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

30  is 

0.8638 

0.5038 

1.7277 

1.0075 

2-5915 

i.5ii3 

3-  4553 

2.0151 

4.3192 

2.5189 

59  45 

30 

8616 

5075 

7233 

0151 

5849 

5226 

4465 

0302 

3081 

5377 

30 

45 

8594 

5ii3 

7188 

0226 

5782 

5339 

4376 

0452 

2970 

5565 

IS 

31  o 

8572 

5150 

7142 

0301 

5715 

5451 

4287 

0602 

2858 

5752 

59  o 

IS 

8549 

5188 

7098 

0375 

5647 

5563 

4196 

0751 

2746 

5939 

45 

30 

8526 

5225 

7053 

0450 

5579 

5675 

4106 

0900 

2632 

6125 

30 

45 

8504 

5262 

7007 

0524 

SSii 

5786 

4014 

1049 

2518 

6311 

15 

32  0 

8480 

5299 

6961 

0598 

5441 

5898 

3922 

H97 

2402 

6496 

58  o 

IS 

8457 

5336 

6915 

0672 

5372 

6008 

3829 

1345 

2286 

6681 

45 

30 

8434 

5373 

6868 

0746 

5302 

6119 

3736 

1492 

2170 

6865 

30 

45 
33  o 

0.8410 
8387 

0.5410 
5446 

1.6821 
6773 

1.0819 
0893 

2.5231 
5160 

1.6229 
6339 

3.3642 
3547 

2.1639 
1786 

4.2052 
1934 

2.7049 
7232 

IS 
57  o 

IS 

8363 

5483 

6726 

0966 

5089 

6449 

3451 

1932 

1814 

7415 

45 

30 

8339 

5519 

6678 

1039 

5017 

6558 

3355 

2077 

1694 

7597 

30 

45 

8315 

5556 

6629 

mi 

4944 

6667 

3259 

2223 

1573 

7779 

15 

34  o 

8290 

5592 

6581 

1184 

4871 

6776 

3162 

2368 

1452 

7960 

56  o 

IS 

8266 

5628 

6532 

1256 

4798 

6884 

3064 

2512 

1329 

8140 

45 

30 

8241 

5664 

6483 

1328 

4724 

6992 

2965 

2656 

1206 

8320 

30 

45 

8216 

5700 

6433 

1400 

4649 

7100 

2866 

2800 

1082 

8500 

IS 

35  o 

8192 

5736 

6383 

1472 

4575 

7207 

2766 

2943 

0958 

8679 

55  o 

15 
30 

0.8166 
8141 

0-5771 
5807 

I  .  1543 
1614 

2.4499 
4423 

I.73I4 
7421 

3.2666 
2565 

2.3086 
.  3228 

4.0832 
0706 

2.8857 
9035 

45 
30 

45 

8116 

5842 

6231 

1685 

4347 

7527 

2463 

3370 

0579 

9212 

15 

36  o 

8090 

5878 

6180 

1756 

4271 

7634 

2361 

35" 

0451 

9389 

54  o 

15 

8064 

5913 

6129 

1826 

4193 

7739 

2258 

3652 

0322 

9565 

45 

30 

8039 

5948 

6077 

1896 

4116 

7845 

2154 

3793 

oi93 

9741 

30 

45 

8013 

5983 

6025 

1966 

4038 

7950 

2050 

3933 

0063 

9916 

IS 

37  o 

7986 

6018 

5973 

2036 

8054 

1945 

4073 

3-9932 

3.0091 

S3  o 

15 

7960 

6053 

5920 

2106 

3880 

8i59 

1840 

4212 

9800 

0265 

45 

30 

7934 

6088 

5867 

2175 

3801 

8263 

1734 

4350 

9668 

0438 

30 

45 

0.7907 

0.6122 

1.5814 

1.2244 

2.3721 

1.8367 

3.1628 

2.4489 

3-9534 

3.o6n 

IS 

38  o 

7880 

6i57 

576o 

2313 

3640 

8470 

1520 

4626 

9400 

0783 

52  0 

IS 

7853 

6191 

57o6 

2382 

356o 

8573 

1413 

4764 

9266 

0955 

45 

30 

7826 

6225 

5652 

2450 

3478 

8675 

1304 

4901 

9130 

1126 

30 

45 

7799 

6259 

5598 

2518 

3397 

8778 

"95 

5037 

8994 

1296 

IS 

39  o 

7771 

6293 

5543 

2586 

3314 

8880 

1086 

5173 

8857 

1466 

Si  o 

IS 

7744 

6327 

5488 

2654 

3232 

8981 

0976 

5308 

8720 

1635 

45 

30 

77i6 

6361 

5432 

2722 

3149 

9082 

0865 

5443 

8581 

1804 

30 

45 

7688 

6394 

5377 

2789 

3065 

9i83 

0754 

5578 

8442 

1972 

IS 

40  o 

7660 

642$ 

5321 

2856 

2981 

9284 

0642 

5712 

8302 

2139 

SO  0 

15 

0.7632 

0.6461 

1.5265 

1.2922 

2.2897 

1.9384 

3-0529 

2.5845 

3.8162 

3.2306 

45 

30 
45 

7604 
7576 

6494 
6528 

5208 
SISI 

2989 
3055 

2812 
2727 

9483 
9583 

0416 
0303 

5978 
6110 

8020 
7878 

2638 

30 
IS 

41  o 

7547 

6561 

5094 

3121 

2641 

9682 

0188 

6242 

7735 

2803 

49  o 

IS 

75i8 

6593 

5037 

3187 

2555 

978o 

0074 

6374 

7592 

2967 

45 

30 

7490 

6626 

4979 

3252 

2469 

9879 

2.9958 

6505 

7448 

3131 

30 

45 

746i 

6659 

4921 

3318 

2382 

9976 

9842 

6635 

7303 

3294 

15 

42  o 

7431 

6691 

4863 

3383 

2294 

2.0074 

9726 

6765 

7157 

3457 

48  o 

15 

7402 

6724 

4804 

3447 

2207 

0171 

9609 

6895 

7011 

3618 

45 

30 

7373 

6756 

4746 

3512 

2118 

0268 

9491 

7024 

6864 

378o 

30 

45 

0.7343 

0.6788 

1.4686 

1.3576 

2.2030 

2.0364 

2.9373 

2.7152 

3.6716 

3-3940 

IS 

43  o 

7314 

6820 

4627 

3640 

1941 

0460 

9254 

7280 

6568 

4100 

47  o 

IS 

7284 

6852 

4567 

3704 

1851 

0555 

9135 

7407 

6419 

4259 

45 

30 

7254 

6884 

4507 

3767 

1761 

0651 

9015 

7534 

6269 

4418 

30 

45 

7224 

6915 

4447 

3830 

1671 

0745 

8895 

7661 

6118 

4576 

IS 

44  o 

7193 

6947 

4387 

3893 

1580 

0840 

8774 

7786 

5967 

4733 

46  o 

IS 

7163 

6978 

4326 

3956 

1489 

0934 

8652 

7912 

58i5 

4890 

45 

30 

7133 

7009 

4265 

4018 

1398 

1027 

8530 

8036 

5663 

5045 

30 

45 

7102 

7040 

4204 

4080 

1306 

1  120 

8407 

8161 

5509 

5201 

IS 

45  o 

7071 

7071 

4142 

4142 

1213 

1213 

8284 

8284 

5355 

5355 

45  0 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

C-/OtU"3C. 

Dist.  I. 

Dist.  2. 

Dist.  3. 

Dist.  4. 

Dist.  5. 

COMPASS   SURVEYING 


147 


TRAVERSE  TABLE  (Continued) 


<rV*iir«» 

Dist.  6. 

Dist.  7. 

Dist.  8. 

Dist.  9. 

Dist.  10. 

V^OuTSc. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

.  30  15 

5.1830 

3.0226 

6.0468 

3.5264 

6.9107 

4-0302 

7-7745 

4-5340 

8.6384 

5-0377 

59  45 

30 

0452 

0314 

5528 

8930 

0603 

7547 

5678 

6163 

0754 

30 

45 

1564 

0678 

0158 

5791 

8753 

0903 

7347 

6016 

5941 

1129 

IS 

31  o 

1430 

0902 

0002 

6053 

8573 

1203 

7145 

6353 

5717 

1504 

59  o 

15 

1295 

1126 

5.9844 

6314 

8393 

1502 

6942 

6690 

5491 

1877 

45 

30 

1158 

1350 

9685 

6575 

8211 

1800 

6738 

7025 

5264 

2250 

30 

45 

1021 

1573 

9525 

6835 

8028 

2097 

6532 

7359 

5035 

2621 

IS 

32  o 

0883 

1795 

9363 

7094 

7844 

2394 

6324 

7693 

4805 

2992 

58  o 

15 

0744 

2017 

9201 

7353 

7658 

2689 

6116 

8025 

4573 

336i 

45 

30 

0603 

2238 

9037 

7611 

7471 

2984 

5905 

8357 

4339 

3730 

30 

45 

5.0462 

3.2458 

5.8S73 

3.7868 

6.7283 

4.3278 

7.5694 

4.8688 

8.4104 

5.4097 

IS 

33  o 

0320 

2678 

8707 

8125 

7094 

3571 

548o 

9018 

3867 

4464 

57  ° 

15 

0177 

2898 

i   8540 

8381 

6903 

3863 

5266 

9346 

3629 

4829 

45 

30 

0033 

3116 

!   8372 

8636 

6711 

4155 

5050 

9674 

3389 

5194 

3° 

45 

4.9888 

3334 

8203 

8890 

6518 

4446 

4832 

S.oooi 

3147 

5557 

IS 

34  o 

9742 

3552 

8033 

9144 

6323 

4735 

4613 

0327 

2904 

5919 

56  o 

15 

9595 

3768 

!  786i 

6127 

5024 

4393 

0652 

2659 

6280 

45 

30 

9448 

3984 

l  7689 

9648 

5930 

5312 

4171 

0977 

2413 

6641 

30 

45 

9299 

4200 

I  7515 

9900 

5732 

5600 

3948 

1300 

2165 

7000 

IS 

35  o 

9149 

4415 

734i 

4.0150 

5532 

5886 

3724 

1622 

1915 

7358 

55  o 

IS 

4.8998 

3.4629 

5.7165 

4.0400 

6.5331 

4.6172 

7.3498 

S.I943 

8.1664 

5-77IS 

45 

30 

8847 

4842 

6988 

0649 

5129 

6456 

3270 

2263 

1412 

8070 

30 

45 

8694 

5055 

6810 

0897 

4926 

6740 

3042 

2582 

1157 

8425 

IS 

36  o 

8541 

5267 

6631 

1145 

4721 

7023 

2812 

2901 

0902 

8779 

54  o 

IS 

8387 

5479 

6451 

45i6 

2580 

3218 

0644 

9UI 

45 

3° 

8231 

5689 

6270 

1638 

4309 

7586 

2347 

3534 

0386 

9482 

3° 

45 

8075 

5899 

6088 

1883 

4100 

7866 

2113 

3849 

0125 

9832 

IS 

37  o 

7918 

6109 

5904 

2127 

3891 

8I4S 

1877 

4163 

7.9864 

6.0182 

53  o 

IS 

7760 

6318 

5720 

2371 

3680 

8424 

1640 

4476 

9600 

0529 

45 

30 

7601 

6526 

5535 

2613 

3468 

8701 

1402 

4789 

9335 

0876 

30 

45 

4-7441 

3.6733 

5.5348 

4.2855 

6-3255 

4.8977 

7.1162 

5.5100 

7.9069 

6.1222 

IS 

38  o 

7281 

6940 

5161 

3096 

3041 

9253 

0921 

5410 

8801 

1566 

52  o 

IS 

7119 

7146 

4972 

3337 

2825 

9528 

0679 

5718 

8532 

1909 

45 

30 

6956 

7351 

4783 

3576 

2609 

9801 

0435 

6026 

8261 

22SI 

30 

45 

6793 

7555 

4592 

3815 

2391 

5-0074 

0190 

6333 

7988 

2592 

IS 

39  o 

6629 

7759 

4400 

4052 

2172 

0346 

6.9943 

6639 

7715 

2932 

51  o 

IS 

6464 

7962 

4207 

4289 

1951 

0616 

9695 

6943 

7439 

3271 

45 

30 

6297 

8165 

4014 

4525 

1730 

0886 

9446 

7247 

7162 

3608 

30 

45 

6131 

8366 

3819 

476l 

ISO? 

"55 

9196 

7550 

6884 

3944 

IS 

40  o 

5963 

8567 

3623 

4995 

1284 

1423 

8944 

7851 

6604 

4279 

50  o 

IS 

4-5794 

3.8767 

5.3426 

4.5229 

6.1059 

5.1690 

6.8691 

5.8151 

7-6323 

6.4612 

45 

30 

5624 

8967 

3228 

546i 

0832 

1956 

8437 

8450 

6041 

4945 

30 

45 

5454 

9166 

3030 

5693 

0605 

2221 

8181 

8748 

5756 

5276 

15 

41  o 

5283 

9364 

2830 

5924 

0377 

2485 

7924 

9045 

5471 

5606 

49  ° 

IS 

5110 

956i 

2629 

6i54 

0147 

2748 

7666 

9341 

SI84 

5935 

45 

30 

4937 

9757 

2427 

6383 

5.9916 

3OIO 

7406 

9636 

4896 

6262 

30 

45 

4763 

9953 

2224 

6612 

9685 

3271 

7145 

9929 

4606 

6588 

IS 

42  o 

4589 

4-0148 

2O2O 

6839 

9452 

3530 

6883 

6.O222 

4314 

6913 

48  o 

IS 

4413 

0342 

I8l5 

7066 

9217 

3789 

6620 

0513 

4022 

7237 

45 

30 

4237 

0535 

1609 

7291 

8982 

4047 

6355 

0803 

3728 

7559 

3° 

45 

4.4059 

4.0728 

5  1403 

4.7516 

5.8746 

5-4304 

6.6089 

6.IO92 

7.3432 

6.7880 

IS 

43  o 

3881 

0920 

1195 

7740 

8508 

4560 

5822 

1380 

3135 

8200 

47  0 

IS 

3702 

IIII 

0986 

7963 

8270 

4815 

1666 

2837 

8518 

45 

30 

3522 

1301 

0776 

8185 

8030 

5068 

5284 

1952 

2537 

8835 

3° 

45 

3342 

1491 

0565 

8406 

7789 

S32I 

5013 

2236 

2236 

9151 

IS 

44  o 

3160 

1680 

0354 

8626 

7547 

5573 

4741 

2519 

1934 

9466 

46  o 

15 

2978 

1867 

0141 

8845 

7304 

5823 

4467 

2801 

1630 

9779 

45 

30 

2795 

2055 

4.9928 

9064 

7060 

6073 

4193 

3O82 

1325 

7.0091 

30 

45 

2611 

2241 

9713 

9281 

6815 

6321 

3917 

3361 

1019 

0401 

IS 

45  o 

2426 

2426 

9497 

9497 

6569 

6569 

3640 

3640 

0711 

0711 

45  o 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dist.  6. 

Dist.  7. 

Dist.  8. 

Dist.  9. 

Dist.  10. 

Course. 

CHAPTER   V 
TRIGONOMETRY 

When  the  surveyor  reads  the  bearing  of  a  compass 
needle  he  reads  an  angle  and  during  his  working  life  he  is 
dealing  with  angular  measurements.  It  is  therefore  neces- 
sary that  he  be  skilled  in  that  special  branch  of  mathematics 
termed  trigonometry. 

Trigonometry  was  formerly  said  to  deal  with  the  prop- 
erties of  triangles  and  was  divided  into  Plane,  Analytical 
and  Spherical  Trigonometry.  It  is  now  defined  as  a  branch 
of  mathematics  dealing  with  the  functions  of  angles  and 
their  application  in  the  solution  of  triangles.  Plane  and 
Analytical  Trigonometry  have  been  merged,  so  we  now  have 
only  Plane  and  Spherical  Trigonometry,  the  latter  merely  a 
particular  application  to  cases  where  the  sides  of  triangles 
are  arcs  of  circles.  It  is  proposed  in  this  chapter  to  give 
the  minimum  amount  of  trigonometry  every  surveyor 
requires. 

Every  triangle  consists  of  six  parts,  three  sides  and  three 
angles.  When  three  parts  are  known,  one  of  which  must 
be  a  side,  the  other  three  can  be  found.  To  the  foregoing 
statement  the  exception  must  be  made  that  when  two  90° 
angles  are  given  with  a  side  opposite  one  of  them  the  solu- 
tion is  indeterminate.  However  the  surveyor  will  not 
meet  with  this  particular  case. 

In  arithmetic  the  three  sides  of  a  right-angled  triangle 
are  known  as  the  base,  the  altitude  and  the  hypothenuse. 

Carpenters  and  practical  men  usually  call  the  three  sides 
the  run,  the  rise  and  the  slope. 

For  convenience  it  is  usual  in  mathematical  work  to 
letter  the  sides  and  angles.  The  acute  angle  at  the  base 
is  known  as  ''Angle  A,"  the  side  opposite  being  marked 
with  a  small  letter  and  called  "side  a."  The  word  " alti- 
tude" begins  with  "a,"  so  the  upright  line  designating  the 

148 


TRIGONOMETRY 


149 


altitude  may  be  easily  remembered  as  "line  a."  Similarly 
the  base  is  "line  b"  and  the  acute  angle  at  the  top  is 
"Angle  B."  The  right  angle  at  the  base  is  "Angle  C," 
the  hypothenuse  opposite  the  angle  being  "line  c. " 


Base 

FIG.  150. 


Run 

FIG.  151. 


FIG.  152. 


The  right-angled  triangle  was  used  for  illustration  but 
the  same  conventions  are  used  for  oblique-angled  triangles; 
i.e.,  capital  letters  indicate  angles  and  small  letters  indicate 
sides. 

THE  PYTHAGOREAN  THEOREM 

Pythagoras,  a  noted  mathematician  of  ancient  times,  is 
said  to  have  been  the  first  to  demonstrate  that 

In  a  right-angled  triangle  the  square  on  the  hypothenuse  is 
equal  to  the  sum  of  the  squares  on  the  other  two  sides,  so  the 
statement  is  known  as  the  Pythagorean  Theorem  and  is 
the  basis  for  a  number  of  trigonometrical  expressions. 

Stated  in  a  modern  way  (algebraically)  it  appears  as 
follows : 


or 

then 

and 


a  =  V(c  +  b)(c-  b)  =  Vc2  -  b\ 


b  =  V(c  +  a)(c-  a)  =  Vc*  -  a2 


GRAPHICAL   SOLUTION   OF   A  RIGHT  TRIANGLE 

Fig.  153  is  reproduced,  with  some  of  the  finer  gradua- 
tions omitted,  from  a  diagram  copyrighted  in  1912  by 
Constantine  K.  Smoley. 


PRACTICAL  SURVEYING 


The  numbers  may  be  taken  to  represent  inches,  feet, 
yards  or  meters.  Given  two  sides  of  any  triangle  the  third 
side  may  be  found. 

Given  the  base  and  altitude  to  find  the  hypothenuse.  —  Find 
the  base  on  A-B  and  the  altitude  on  B-C.  Follow  the 


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i 

FIG.  153.     Graphical  solution  of  a  right  triangle. 

lines  to  an  intersection  and  then  trace  the  arc  found  at  this 
point  to  line  A— D,  on  which  read  the  length  of  the  hypothe- 
nuse. 

Given  the  hypothenuse  and  another  side  to  find  the  third 
side.  —  Place  a  ruler  on  the  line  marking  the  second  side. 
If  this  side  is  the  altitude  the  ruler  will  be  parallel  with 
A-B,  starting  from  B-C.  If  the  second  side  is  the  base 
the  ruler  will  be  parallel  with  B— C,  starting  from  A-B. 
Find  the  hypothenuse  on  A-D  and  follow  the  arc  inter- 
secting A-D  at  this  point  to  an  intersection  with  the  ruler. 
Going  from  this  intersection  on  a  line  perpendicular  to 
the  ruler  read  the  required  third  side. 

The  diagram  presented  appears  in  the  seventh  edition 
of  Smoley's  Tables,  the  standard  tables  for  engineering 
draftsmen,  and  is  in  inches.  The  author  used  similar 
diagrams  for  many  years  drawn  on  cross-section  paper 


TRIGONOMETRY 


divided  decimally  instead  of  in  eighths.  Such  diagrams  are 
in  fairly  common  use  and  are  of  considerable  value  in  cer- 
tain kinds  of  work.  For  instructional  purposes  they  are 
very  good. 

TRIGONOMETRIC   FUNCTIONS 

In  trigonometry  all  angles  are  assumed  to  have  angle  A 
at  the  center  of  a  circle,  side  b  and  side  c  being  radial  lines 
intercepting  an  arc  on  the  circumference.  Either  b  or  c 
may  be  a  radius. 

If  an  arc  is  described  and  a  perpendicular  let  fall  from 
the  end  of  the  radius  to  the  base,  then 


a  =  sine, 
b  =  cosine, 
c  =  radius, 
c  —  b  =  versed  sine, 


written  sin, 
written  cos, 
written  rod, 
written  versin. 


b-co-slne         C 

FIG.  154. 


b-radius 

FIG.  155. 


If  an  arc  is  described  and  a  perpendicular  erected  from 
the  end  of  the  radius  as  the  base,  then 

a  =  tangent,  written  tan  or  tangt, 
b  =  radius,     written  rod, 
c  =  secant,     written  sec. 

The  sine,  tangent,  etc.,  are  called  functions  of  an  angle, 
a  function  in  mathematics  being  any  algebraic  expression 
or  quantity  dependent  for  its  value  on  another  one. 

The  difference  between  any  angle  in  a  quadrant  and  90 
degrees,  the  full  quadrantal  angle,  is  said  to  be  the  comple- 
ment of  the  angle.  Thus  in  Fig.  156,  angle  B  is  the  com- 
plement of  angle  A  and  vice  versa. 


152 


PRACTICAL  SURVEYING 


-Cotangent  


The  functions  of  the  complement  are  said  to  be  co-func- 
tions of  the  angle. 

The  sine  of  A  —  cosine  of  B. 

The  sine  of  B  =  cosine  of  A . 

The  tangent  of  A  =  cotangent  of  B. 

The  cotangent  of  B  =  tangent  of 
A,  etc.,  as  shown  in  Fig.  156. 

The  difference  between  any  angle 
and  1 80  degrees  is  called  the  supple- 
ment of  the  angle.  The  functions 
of  the  supplement  are  the  functions 
of  the  angle. 


£• —  Cosine-— r1 
FIG.  156.     Functions  of 
angles. 


Side  a  may  be  the  sine  or  tangent  of  angle  A  and  side  b 
may  be  the  cosine,  or  it  may  be  the  radius  of  the  circle  in 
which  a  triangle  is  drawn  having  angle  A  at  the  center. 

TO  avoid  confusion  it  is  considered  best  to  regard  the 
functions  as  ratios,  that  is,  as  pure  numbers  and  not  as 
lines.  Side  a  and  side  b  are  lines  but  sin 
A  ,  tan  A  and  cos  A  are  not  always  lines. 
Tangent  A  does  not  become  a  line  un- 
til it  is  multiplied  by  the  run  of  the  tri- 
angle having  angle  A  at  the  base,  when 
it  becomes  side  a.     If  sin  A  is  multi- 
plied by  th$  slope  it  becomes  side  a. 

Using  the  "ratio  concept"  the  functions 
as  follows,  referring  to  Fig.  157. 

Sin  A  =-• 
c 


FIG.  157. 

are  described 


Cot  A  = 


Sec  A  =T 


Cosec  A  =  -- 
a 


TRIGONOMETRY  153 

The  relations  (ratios)  here  shown  are  true,  no  matter 
what  may  be  the  lengths  of  the  lines.  The  ratios  must  be 
memorized  or  the  student  will  find  himself  as  helpless  in 
trigonometrical  work  as  he  would  be  in  arithmetic  with- 
out knowing  the  multiplication  tables. 

The  following  trigonometrical  equivalents  must  also  be 
memorized,  an  easy  task  when  the  connection  with  the 
Pythagorean  theorem  is  noted. 


cos 


Sin  =       -  =  -    =  V(i  -  cos2)  =  V(i+cos)(i  -cos), 
cosec     cot 

Cos  =  ^  =  —  =  sin  X  cot  =  V(i  -  sin2) 
tan      sec 

=  V(i  -+-  sin)  (i  —  sin). 

^          sin        I 

Tan  =  -~  =  —  • 
cos      cot 

cos        i 
Cot  =  —  —--' 
sin      tan 

Sec  =  t^  =  —  =  Vrad2  +  tan2. 
sin      cos 

Cosec  =  — — 
sin 


Rad  =  tan  X  cot  =  Vsin2  -f  cos2, 
c-b 


Versin  =  rod  —  cos  = 
Coversin  =  rod  —  sin  = 


c 
c  —  a 


NUMERICAL  VALUES  FOR  THE  TRIGONOMETRICAL  RATIOS 


Angle  o 

Draw  a  square  A,  B,  C,  D  and  in  it  draw  the  diagonal 


° 


00 
A,  B.     Angle  x  =  45°  =  -  —  because  side  a  =  side  b. 

c2  =  a2  +  b2, 
and  since  a  =  b  =  I, 

c2  =  2b2  =  2a2;    or    c  =  \/2  X  b  =  \/2  X  a. 


154 


PRACTICAL  SURVEYING 


.o      b        i 


Cos 45'  ----i-. 


Tan  45°-? -I. 


FIG.  158.    Functions  of  45°.      FIG.  159.    Functions  of  30°  and  60°. 

Angle  of  60°. 

In  the  equilateral  triangle  (Fig.  159)  each  of  the  three 
angles  is  equal  to  60°.  Drop  the  perpendicular  BD  to  b. 
Then,  assuming  the  length  of  each  side  =  I, 


that  is, 


2  AD  =  AB  =  AC  =  BC  =  a  =  b  =  c  =  I, 
=  BD2+  AD2  =  a2=b2  =  (2AD)2  =  4  AD* 
4  AD2  =  BD2  +  AD2    or     BD2  =  3  AD2; 


4  X  i2  =  BD2  +  i2     or 
...     BD  =  V3  X  i  = 


3  X  i. 


Since  angle    ^4  =  6oc 


Cos  60°  = 


AD 


I         2 


TRIGONOMETRY 


155 


Angle  of  30°. 

Sin  30°  =  cos  60°  =  J. 

V\ 
Cos  30°  =  sin  60°  =  —^ 

Tan  30°  =  cot  60°  = 


tan  60 


Sin  15°  = 


Sin  i 8°  = 


_ 

V2 


To  obtain  the  remaining  functions  of  the  foregoing  angles 
refer  to  the  list  of  Trigonometrical  Equivalents. 

SIGNS   OF  THE   TRIGONOMETRICAL  RATIOS 

It  is  convenient  (hence  the  word  convention)  to  assume 
that  all  directions  up  or  down  are  measured  from  a  hori- 
zontal line  (or  axis)  X  .  .  .  X'  and 
that  all  directions  right  or  left  are 
measured  from  a  vertical  line  (or  axis) 
Y  .  .  .  Yf .  The  vertical  axis  is  termed 
an  ordinate  and  the  horizontal  axis  an 
abscissa.  The  axes  are  known  as  co- 
ordinates and  the  point  of  intersection, 
0,  the  origin  of  co-ordinates. 

Describing  a  circle  with  O  as  a  center 
and  O  ...  X  as  a  radius  the  axes  di- 
vide it  into  four  quarters,  or  quadrants. 
Proceeding  around  the  circle  in  a  direction  contrary  to 
that  followed  by  the  hands  of  a  clock  —  anti-clockwise  — 
the  quadrants  are  numbered  as  shown  in  Fig.  160. 

All  motion  toward  X  (to  the  right)  and  towards  Y  (up) 
is  considered  positive  (  +  ).  Motion  to  the  left  and  down  is 
considered  negative  (  — ). 

The  sine  and  cosine  cannot  extend  beyond  the  circum- 
ference of  a  circle  having  a  radius  =  I  but  the  tangent, 
cotangent,  secant  and  cosecant  may  extend  to  infinity  (<*>). 


156 


PRACTICAL  SURVEYING 


Representing  the  functions  by  lines,  the  signs  of  the  func- 
tions, together  with  their  limiting  values,  are  shown  in  the 
following  table,  x  being  the  angle  at  the  origin. 


If  the  angle  is  in 
quadrant. 

Sin*. 

Cos*. 

Tan*. 

Cot  *. 

Sec*. 

Cosec  *. 

T   (Sign  

+ 

+ 

+ 

+ 

+ 

+ 

H  Value... 

o  to  i 

I  tO  0 

o  to  oo 

oo  to  o 

I  tO  00 

00  tO  I 

TT  jSign  
1L  IValue... 

i  to  o 

o  to  i 

00  tO  O 

o  to  oo 

oo  to  i 

i  to  oo 

TTT   (Sign  
[IL  IValue  ... 

o  to  i 

I  tO  0 

o  to  oo 

oo  to  o 

I   tO  CO 

00  tO  I 

IV  (  Sign  



_l_ 





+ 



*  1  Value  .  .  . 

i  to  o 

o  to  i 

oo  to  o 

o  to  oo 

oo  to  i 

I   tO  00 

Summarizing  the  results  of  the  investigation  of  the  nu- 
merical values  the  following  table  has  been  prepared: 


^^^ 

o°or 
360° 

30° 

45° 

60° 

90° 

180° 

270° 

Sine                      .   .  . 

o 

jj 

I 

V3 

o 

I 

V2 

2 

Cosine              

i 

^1 

I 

\ 

o 

—  I 

o 

2 

V2 

I 

Tangent  

o 

/— 

I 

•\/3 

00 

0 

00 

V3 

Sin  30°  =  J  =  0.5  =  cos  60°. 

Sin  45°  =  4=  =  -     -  =  0.7071  =  cos  45° 


I4I4 


Sin  60°  = 
Tan  30°  = 
Tan  60°  = 


=  0.866  =  cos  30°. 
=  °-5774  =  cot  60°. 


1.732  = 


TRIGONOMETRY  157 

TABLE  OF  NATURAL  FUNCTIONS 

The  circumference  of  a  circle  is  divided  into  360  equal 
parts  called  degrees. 

Each  degree  is  divided  into  60  equal  parts  called  minutes. 

Each  minute  is  divided  into  60  equal  parts  called  seconds. 

The  size  of  an  angle  is  indicated  by 
the  number  of  parts  of  a  circle  contained 
in  the  intercepted  arc;  thus,  25°  15'  10" 
which  is  read  25  degrees,  15  minutes,  10 
seconds. 

In  Fig.  161  the  angle  at  A  =  30°, 
at  C  -  90°  and  at  B  =  60°.     The  line 
A-B  is  75.3  ft.  long.     What  are  the  lengths  of  the  other 
sides? 

Referring  to  the  list  of  Trigonometrical  Equivalents 

c         tan 
Sec  =  -^-i 
sin 

therefore 

sin  X  sec  =  tan. 

In  the  30°  triangle  the  sin  =  0.5  and  as  the  slope  has  a 
length  of  75.3  ft.,  the  tangent  (line  B-C)  has  a  length  of 
0.5  X  75-3  =  37.65  ft. 

Also  the  secant  =  —  > 
cos 

therefore  cos  X  sec  =  I  =  radius. 

The  cos  of  30°  =  0.866  and  the  length  of  A-C  =  75.3  X 
0.866  =  65.21  ft. 

Use  of  tangent. 

(i)  A  triangle  has  a  base  of  65.21  ft.  and  an  altitude  of 
37.65  ft.  What  is  the  angle  A  ? 

sin       alt.      a      37. 65 


The  tangent  for  an  angle  of  30°  =  0.5774  so  the  angle 
A  =  30°. 

(2)  A  30°  right-angled  triangle  has  a  base  of  65.21  ft. 
What  is  the  altitude? 


-158  PRACTICAL  SURVEYING 

The  tangent  of  30°  =  0.5774  so  the  altitude  =  0.5774  X 
65.21  =  37.65  ft. 

By  methods  given  in  college  textbooks,  tables  have  been 
computed  which  contain  values  of  all  the  functions.  So 
many  tables  are  in  existence  it  is  unnecessary  for  any  one 
save  professional  mathematicians,  as  a  part  of  their  train- 
ing, to  compute  such  tables. 

The  table  of  Natural  Functions  of  Angles  here  presented 
gives  values  of  the  ratios  for  each  ten  minutes  of  arc,  this 
being  sufficient  for  the  examples  in  this  book,  which  are 
intended  to  give  practice  in  the  solution  of  triangles  and 
the  use  of  tables.  Tables  in  common  use  give  values  for 
each  minute  of  arc,  with  numbers  whereby  interpolations 
may  be  made  for  smaller  angles.  Books  of  tables  usually 
contain  directions  for  use. 

Example.  —  Find  the  sine  of  33°  20'. 

Turning  to  the  table  find  33  in  the  column  headed  (Deg.) 
and  find  20  in  the  column  headed  (Min.).  The  sine  is  in 
the  column  headed  Sine  and  =  0.564007.  The  radius  has 
a  value  =  i.oooooo. 

The  values  of  other  functions  are  found  in  the  same  way 
in  the  proper  column. 

Example.  —  Find  the  tangent  of  54°  30'. 

The  tangent  of  54°  30'  =  1.4019483. 

Read  the  note  at  the  bottom  of  each  page  of  the  table. 
The  functions  of  angles  are  co-functions  of  the  complements 
of  the  angles  so  it  is  therefore  necessary  to  compute  the 
values  of  functions  for  angles  between  o°  and  45°  only. 
For  angles  between  45°  and  90°  read  up  from  the  bottom 
of  the  pages.  Notice  that  the  column  with 

Sine  at  the  top  has  Cosine  at  the  bottom, 
Tangent  at  the  top  has  Cotangent  at  the  bottom, 
Secant  at  the  top  has  Cosecant  at  the  bottom 
and  vice  versa. 

Example.  —  Find  the  sine  of  146°  40'. 
Subtract  from  179°  60'  (180°) 

146°  40' 
and  find  sine  of  33°  20' 

Example.  —  Find  the  tangent  of  125°  30'. 


TRIGONOMETRY  159 

Subtract  from  179°  60'  (180°) 

125°  3Q' 
and  find  tangent  of  54°  30' 

These  two  examples  show  that  the  functions  of  an  angle 
are  the  functions  of  the  supplement  of  the  angle. 

In  adding  and  subtracting  angles  it  is  a  good  safe  habit 
to  write  the  work  on  paper  instead  of  attempting  to  do  it 
mentally  until  considerable  experience  is  had.  It  is  well 
to  see  such  things. 

In  subtraction  lessen  the  degrees  by  I  and  add  60  to  the 
minutes  of  the  larger  angle.  When  seconds  are  given  lessen 
the  number  of  minutes  by  I  and  add  60  to  the  seconds. 

The  tables  in  this  book  give  values  to  10  minutes  of  arc 
but  for  all  practical  purposes  the  values  for  intermediate 
angles  may  be  obtained  by  interpolation. 

Example.  —  Find  sine  of  27°  13'. 

Sine  27°  20'  =  0.459166 
Sine  27°  10'  =  0.456580 

0.002586  diff.  for  10' 
3  minutes  =  0.3  of  10  minutes,  therefore 
0.002586  diff.  for  10' 

0-3 

0.0007758 
Sine  27°  10'  04565800 

04573558  =  sin27°!3/. 

In  using  the  tables  in  this  book  for  arcs  smaller  than  10 
minutes,  the  final  results  should  be  considered  as  being 
correct  for  the  first  five  places  (figures).  If  the  sixth  figure 
is  5  or  more,  increase  the  fifth  figure  by  I  and  reject  the 
figures  following.  Thus,  the  sine  of  27°  13'  correct  to  five 
places  =  045736.  If  the  sixth  figure  is  less  than  5  drop  all 
figures  following  the  fifth. 

GRAPHICAL  NATURAL  FUNCTIONS 

Paper  protractors  8  and  14  ins.  in  diameter  are  printed 
from  engine  divided  plates  on  rectangular  sheets.  On 
such  a  sheet  draw  a  line  through  the  center  passing  through 
the  270°  and  90°  graduations.  Normal  to  this  line  draw 


160  PRACTICAL  SURVEYING 

a  line  from  the  center  through  the  o°  point.  Make  each 
line  5  or  10  ins.  long  and  divide  in  ten  equal  parts.  Draw 
lines  through  each  division  so  the  paper  will  be  ruled  in 
squares  which  are  to  be  numbered  from  o  to  10,  starting 
at  the  center  of  the  protractor.  Each  division  should  be 
divided  into  ten  parts.  A  fine  line  drawn  from  the  center 
through  any  angle  will  represent  the  hypothenuse  of  a 
triangle.  This  hypothenuse  is  the  secant  of  the  angle  with 
a  radius  =  i.o.  In  making  practical  use  of  such  a  diagram 
a  fine  line  is  drawn  on  a  narrow  strip  of  tracing  cloth  or 
transparent  celluloid  and  this  line  is  graduated  to  corre- 
spond with  the  horizontal  and  vertical  lines.  A  fine  needle 
is  put  through  the  zero  on  this  line  and  the  center  of  the 
protractor.  The  transparent  strip  may  then  be  swung  so 
the  line  will  intersect  any  angle  and  the  lengths, of  the  three 
sides  of  a  triangle  having  this  angle  at  the  base  may  be 
read. 

If  the  angle  A  alone  is  given  the  secant  may  be  extended 
an  infinite  distance  and  an  infinite  number  of  triangles  be 
formed  in  which 

B  =  90° -4, 
C  =  90°, 

for  the  sum  of  the  three  interior  angles  of  any  triangle  = 
2  x  90  =  1 80°. 

An  angle  measures  the  amount  of  divergence  between  two 
lines  starting  from  a  common  point.  The  base  is  one  line 
and  the  secant  another  line  defining  an  angle.  When  the 
secant  is  defined  on  the  diagram  as  described  and  the  angle 
A  thus  marked,  the  angle  B  may  also  be  read  off,  for  it  is 
the  complement  of  the  angle  at  the  base.  To  assist  in 
obtaining  the  complementary  angle  a  second  set  of  gradu- 
ations may  be  placed  on  the  circumference  of  the  protractor 
in  an  opposite  direction  to  the  regular  marks.  This  second 
set  should  be  in  red  ink  to  avoid  mistakes. 

The  radius  being  1 .00  the  functions  of  an  angle  are  ratios 
expressed  in  per  cent  of  the  radius. 

If  the  radius  is  ten  inches,  the  heavy  lines  drawn  at 
intervals  of  one  inch  and  the  lighter  lines  at  intervals  of 
one- tenth  of  an  inch,  values  of  the  functions  may  be  read 
to  three  decimal  places. 


TRIGONOMETRY 


161 


Fig.  162  is  an  engine-divided  diagram  used  in  Lewis 
Institute,  Chicago,  111.,  and  reproduced  by  permission  of 
Prof.  Phillip  B.  Wood  worth.  The  only  difference  between 
this  engine-divided  protractor  and  the  diagram  just  de- 
scribed is  in  the  angular  graduations,  which  are  placed  on 
vertical  and  horizontal  lines  outside  of  the  quadrille  ruling. 


EH 

5'      &>'      75"       ft 

vvvyv^vv^^ 

V    ' 

45y 
/^ 

3     | 

35* 
7    <: 

Aftf". 

--^ 

-^ 

^ 

V 

x 

x 

\ 

x 

c 
a 

CJi 

c 

\ 

. 

\ 

x 

x 

\ 

x 

X1 

\ 

X 

i 

>^ 

A^ 

^ 

CO 

\ 

. 

/ 

s 

> 

x 

, 

' 

2 

Cosine.  6  , 

FIG.  162.     Trigonometer. 

Arranged  as  shown  in  the  illustration  this  device  is  called 
a  Trigonometer  and  is  of  considerable  value  in  checking 
calculations  for  latitudes  and  departures.  Metal  trigo- 
nometers  were  formerly  sold  by  instrument  dealers  but  the 
ease  with  which  a  good  draftsman  may  make  one  probably 
caused  the  manufacture  to  be  discontinued,  for  they  seem- 
ingly are  no  longer  advertised.  The  writer  used  one  for 
many  years. 

SOLUTIONS  OF  RIGHT  TRIANGLE 

Given  A  and  c  to  find  B,  a  and  b. 
B  =  90°  -A, 
a  —  c  sin  A , 
b  =  c  cos  A . 


D 

FIG.  163. 


1 62  PRACTICAL  SURVEYING 

Given  A  and  a,  to  find  B,  b  and  c. 
B  =  90°  -A, 
b  =  a  cot  A , 

a 

sin  A 

Given  A  and  b,  to  find  B,  a  and  c. 
5  =  90°  -A, 
a  =  b  tan  ^4 , 


cos  A 
Given  c  and  a,  to  find  A ,  £  and  b. 

Sin  ^4  =  -  » 
c 

5  =  90°-^, 
6  =  a  cot  ^4 . 

Given  a  and  6,  to  find  A,  B  and  c. 

Tan  A  =  -7  > 

D 

B  =  90°  -4, 

a 


c  = 


sn  ^ 

A  ab 

Area  =  —  • 

Either  acute  angle  may  be  assumed  to  be  the  angle  at 
the  base  in  which  case  the  adjacent  side  becomes  the  base. 

PROBLEMS 

1.  A  =  48°  if,  c  =  324  ft.       Find  B,  a,  b,  area. 

2.  A  =  51°  19',  b  =  1254  ft.     Find  B,  a,  c,  area. 

3.  A  =  43°  38',  a  =  1 86  ft.       Find  B,  b,  c,  area. 

4.  a  =  249  ft.,  c  =  415  ft.        Find  A,  B,  b,  area. 

5.  a  =  67,  b  =  53.         Find  A,  B,  c,  area. 

6.  c  =  893,  b  =  586.     Find  ^4,  5,  a,  area. 

7.  yl  =  64°  40',  &  =  326.     Find  B,  a,  c,  area. 

8.  A  =  71°  24',  a  =  286.     Find  B,  6,  c,  area. 

9.  ^4  =  41°  48',  c  =  963.      Find  5,  b,  a,  area. 


TRIGONOMETRY  163 

In  solving  problems  the  student  is  greatly  assisted  by 
drawing  the  triangle  free  hand  and  writing  the  given  values 
in  the  proper  places.  Each  part  as  found  should  be  placed 
on  the  sketch.  Check.  —  Draw  the  triangles  to  scale,  using 
a  protractor  to  measure  the  angles. 

The  expression  abc  means  a  X  b  X  c,  the  multiplication 
sign  being  understood  when  letters  are  used.  Sin  A  means 
the  sine  of  the  angle  A  ,  and  b  sin  C  means  side  b  X  sin  C, 
the  value  of  the  sine  being  given  in  the  tables. 

Sine  J  A  means  the  sine  of  one-half  the  angle  A  and  does 
not  mean  half  the  sine  of  A,  which  is  something  entirely 
different.  Beginners  often  get  into  trouble  over  this  matter. 

ALGEBRAIC   THEOREMS 

I.  The  square  of  the  sum  of  two  quantities  is  equal  to  the 
square  of  the  first,  plus  twice  the  product  of  the  first  multiplied 
by  the  second,  plus  the  square  of  the  second. 

Example.  —  (a  -f  b)2  =  a2  +  2  ab  -f  b2. 
a  +b 
a  +b 


+  ab  + 


a2  +  2  ab  +  b2 

2.  The  square  of  the  difference  of  two  quantities  is  equal  to 
the  square  of  the  first  minus  twice  the  product  of  the  first  by 
the  second,  plus  the  square  of  the  second. 

Example.  —  (a  -  b)2  =  a2  -  2  ab  +  b2. 
a  -b 
a  -b 
a2  —  ab 
-ab  +  b2 


a2  -  2  ab  +  b2 

3.    The  product  of  the  sum  and  difference  of  two  quantities 
is  equal  to  the  difference  of  their  squares. 


164  PRACTICAL  SURVEYING 

Example.  —  (a  +  b)  (a  -  b)  =  a2  -  b2. 
a  +b 
a    -b 
a2  +  ab 
-ab-b* 


(Note.  —  In  the  preceding  chapter  it  was  explained  that 
in  multiplication  like  signs  produce  +  and  unlike  signs 
produce  — .) 

TRIGONOMETRIC   LAWS 

The  following  laws  should  be  thoroughly  understood  and 
in  their  development  appears  the  Pythagorean  Theorem 
and  the  algebraic  rules  and  theorems  just  given.  The 


C  D  A      A  b  C      A 

FIG.  164. 

student  should  work  the  expressions  by  assuming  the  fol- 
lowing values,  a  =  9,  b  =  12,  c  =  15. 
Law  of  sines. 

In  any  triangle  the  sides  are  to  one  another  as  the  sines 
of  their  opposite  angles. 

a  _  sin  A     b  _  sin  B     a  __  sin  A 
b      sin  B  '  c      sin  C  '  c      sin  C 

Law  of  cosines. 

a2  =  b2  +  c2  —  2  be  cos  A . 

tf  =  a2  +  c2  -  2accosB. 

c*  =  a2  -f  b2  -  2  ab  cos  c. 
Law  of  tangents. 

a  -  b  =  tan  \  (A  -  B) 

a  +  b  ~  tan  \  (A  +  B) ' 

a  -  c  =  tan  \  (A  -  C) 

a  +  c  ~  tan  |  (A  +  C) ' 

b  -  c  _  tan  %(B  -  C) 


TRIGONOMETRY  165 

Half  the  difference  of  two  unequal  quantities  AB  and  BC, 
added  to  half  their  sum,  gives  the  greater,  and  half  the  differ- 
ence taken  from  half  the  sum, 
gives  the  less.  * £ 

Proof.- Draw  AB  +  BC.  '^^ 

Make  AD  =  BC. 

Then  A  C  =  their  sum  and 
BD  =  their  difference. 

Bisect  BD  in  E. 

Then  BE  =  ED  =  half  their  difference  and 
AE  =  EC  =  half  their  sum. 

Consequently  AE  +  EB  =  AB,  the  greater  and 
EC  -  EB  =  BC,  the  less. 

Also,  half  the  difference  BE,  added  to  the  less  BC,  or 
taken  from  the  greater  AB,  gives  half  the  sum. 


SOLUTION   OF  OBLIQUE  TRIANGLES 
FIRST  CASE. 
Given  A,  B,  a,  to  find  C,  6,  c. 

C=  180°-  (A  +3).  (i) 


FIG.  166. 


c  =  sin  C-  —  7-  (3) 

sin  A 

SECOND  CASE. 

Given  A,  a,  b,  to  find  B,  C,  c. 


(4) 


Cuse  (i). 
c  use  (3). 


l66  PRACTICAL  SURVEYING 

THIRD  CASE. 

Given  A,  b,  c,  to  find  a,  B,  C,  area. 

a  =  v62  -f  c2  —  2  be  cos  yl  .  (5) 

B  use  (4). 
C  use  (i). 

a  =  sin  A  —  —  ^;     or  sin  A—  —  ^-  (6) 

sin  B  sin  C 


. 

Area  =  --  -  --  (7) 

The  second  case  has  an  angle  at  the  base  with  the  base 
and  side  opposite  the  angle  given. 

The  third  case  has  an  angle  and  the  two  including  sides 
given. 

The  second  case  often  presents  difficulties  to  the  student. 
If  the  given  angle  is  acute  and  the  side  opposite  is  the  lesser 
side,  then  the  angle  found  may  be  either  obtuse  (greater 
than  90°),  or  acute  (less  than  90°).  The  conditions  of  the 
problem  should  be  known. 

The  sign  <  means.  "less  than"  and  the  sign  >  means 
"greater  than."  The  sign  =  means  "equal  to  or  greater 
than." 

Let  6,  c,  B  be  the  parts  known.  If  b  <  c  sin  B,  no 
solution  is  possible;  if  b  =  c  sin  B,  then  sin  C  =  90°;  if 
b  >  c  sin  B  and  b  <  c,  and  B  is  acute,  two  solutions  are 
possible,  but  if  b  =  c  only  one  solution  is  possible  and  C  is 
an  acute  angle. 

Professor  Stang  in  Engineering  News,  Dec.  25,  1913, 
presented  the  following  formula  for  solving  a  triangle  in 
which  is  given  two  sides  and  the  included  angle: 

cotB  =       a   _  -  cotC. 
osin  C 

FOURTH  CASE. 

Given  a,  b,  c,  the  three  sides,  to  find  A,  B,  C,  the  three 
angles. 

Let 


SinM   =V/^  -^—  • 


TRIGONOMETRY  167 


B  use  4. 
Cuse  I. 


Area  =  Vs  (s  -  a)  (s  -  b)  (s  -  c).  (9) 

The  following  formulas  may  also  be  used  for  this  case. 


-v/ 


Co.SB-V^*^2-  Tania-V/''"*^'"'' 


Cos     C  =  l.  Tan     C  = 


s  (s  —  c) 


The  following  item  appeared  in  Engineering  News,  July 
25,  1912. 

A  triangle  formula,  to  obtain  the  angles  when  the  three 
sides  are  known,  is  given  by  Prof.  C.  Frank  Allen  (Mass. 
Inst.  of  Technology).  While  not  new  it  may  be  unknown 
to  many  of  our  readers,  as  the  formula 


t  l( 

=V 


(s  -  b)  (s  -  c) 


is  usually  quoted  without  reference  to  the  other  method  of 
solution.  In  the  above  the  sides  are  a,  b  and  c,  with  the 
opposite  angles  A  ,  B  and  C,  and  s  is  half  the  sum  of  the  sides. 
The  derivation  and  formula,  given  by  Professor  Allen, 
are  as  follows  :  From  angle  C  drop  a  perpendicular  on  side 
c,  dividing  c  into  the  segments  k  and  g,  adjacent  respec- 
tively to  sides  a  and  b.  Calling  the  length  of  the  perpen- 
dicular h,  we  have: 

w  =  p  -  g2  =  b2  -  (c  -  k)2, 

and 

h*  =  a2-  k2. 
Then 


1  68  PRACTICAL  SURVEYING 

Whence 

and 


(2) 


Since  g  =  c  —  k,  we  get 


When  g  and  &  are  computed,  the  angles  are  readily  found, 
as 

n  k 

cos  A  =  |  »  cos  B  =  -• 

b  a 

For  many  uses  these  formulas  are  more  convenient  than 
Eq.  (i),  being  probably  less  liable  to  errors  of  computation, 
and  more  quickly  derived  if  forgotten. 

PROBLEMS 

In  the  following  problems  give  both  solutions  when  two 
are  possible.  Find  in  each  problem  all  the  parts  not  given 
and  the  area.  Check  the  work  by  drawing  to  scale  and 
read  angles  with  a  protractor. 

1.  c  =  532,  b  =  358,  C  =  107°  40'. 

2.  c  =  232,  b  =  345,  C  =  37°  20'. 

3.  b  =  560,  a  =  258,  B  =  63°  28'. 

4.  B  =  63°  48',  A  =  49°  25',  b  =  275. 

5.  A  =  49°  25',  C  =  63°  48',  6  =  275. 

6.  A  ship  sailing  due  north  observes  a  cape  bearing 
N  54°  12'  W    and  after  sailing  27  miles,  the   cape   bore 
5  70°  30'  W.     Required  her  distances  from  it. 

7.  c  =  133,  b  =  176,  A  =  73°  16'. 

8.  c  =  237,  a-  =  482,  B  =  77°  48'. 

9.  &  =  78,  a  =  168,  C  =  128°  26'. 

10.  a  =  230,  b  =  365,  c  =  426. 

11.  a  =  1248,  b  =  728,  c  =  956. 

12.  a  =  375,  b  =  275,  c  =  196. 


TRIGONOMETRY 


169 


FUNCTIONS   OF  HALF  AN  ANGLE 
In  Fig.  167  let  0,  0,  c  be  the  angle  z,  then 

b,  o,f  =  c,  0,/  =  |z, 
sin  c,  0,  /  =  sin  \  z  =  c  .  .  .  g, 
tan  c,  0,  /  =  tan  \  z  =  e  .  .  .  /, 
sin  z  =  a  .  .  .  c, 
tan  z  =  b  .      .  d. 


A  study  of  this  figure  will  show  why 
the  student  must  be  careful  to  note  the    FIG.  167.  Functions 
difference  between  \  sin  A  and  sin  \  A .         of  half  angles. 


Sinjz  = 


Cosiz  = 


—  cosz 


—  cosz 
H-  cosz 
+  cosz 


-+-  cosz 
—  cosz 

Work  the  above  expressions  with  z  =  45°. 
The  sign  ±  means  "positive  or  negative." 

CIRCULAR   MEASURE  OF  AN  ANGLE 

In  certain  kinds  of  calculations  the  radian  is  the  unit  of 
measure  when  the  magnitude  of  an  angle  is  to  be  measured. 
In  Fig.  1 68  the  arc  ab  is  equal  in  length 
to  the  radius  oa. 


The  ratio 


arc 


is  called  the  circular  or 


radius 
radian  measure  of  an  angle. 

The   circumference    of   a   circle  =  ird,   in 
which  TT  =  3.1416  (pronounced  pi)  and  d  = 
diameter  =  2  X  radius. 
The  circumference  of  a  circle  of  radius  r  is  2  wr,  or,  if  the 
radius  is  unity,  2  w. 


FlG  radfan 


170 


PRACTICAL   SURVEYING 


The  angle  360°  corresponds  to  an  arc  with  the  length  2  TT; 
the  angle  180°  to  an  arc  =  ?r;  the  angle  of  90°  to  an  arc  = 
JTT;  etc. 

The  radian  has  a  value  when  expressed  in  degrees  as 
follows : 

360°  =  1 80° 

27T      "  7T 


i8oc 


3.I4I6 


=  57°  if  44.8"  =  57-295°. 


The  value  usually  given  is  57.3°. 

The  radian,  or  57.3  rule,  is  convenient  when  the  size  of 
angle  between  a  random  line  and  true  line  is  wanted.  If 
the  angle  is  found  to  be  less  than  6°  the  rule  is  all  right, 
but  for  larger  angles  may  introduce  a  considerable  error. 
This  is  because  the  offset  is  always  measured  on  a  straight 

line  and  in  the  57.3  rule  it 

is   assumed    to   be   an    arc. 

In    Fig.    169    this    is    illus- 

trated. 

The  line  AC  is  measured 

on  the  supposition  that  it  is 
the  line  AB,  which  it  however  misses  at  B  by  the  distance 
BC.  For  a  small  angle  the  difference  between  BC  straight 
and  curved  is  so  small  that 


Angle  BAG 


FIG.  169.     Application  of  57.3  rule. 


A  C 


distance 


In  higher  mathematics  the  radian  measure  of  an  angle  is 
necessary  but  the  instance  just  given  is  the  only  one  in 
which  the  surveyor  can  advantageously  use  it. 


HEIGHTS  AND   DISTANCES 

In  Fig.  170,  DA  is  parallel  to  BC  and 
the  angle  ABC  =  90°. 


47°  25'. 


FIG.  170. 


Find  height  AB. 

2.  J3C=i36ft.,  angle  .4  C5 
Find  height  AB. 

3.  The  angular  elevation  of  a  wall,  taken  from  the  edge 
of  a  ditch  18  ft.  wide,  was  62°  40'.     Required  the  height 
of  the  wall  and  the  length  of  a  ladder  to  reach  the  top  of  it. 


TRIGONOMETRY  171 

4.    Let  the  sloping  side  of  a  hill  AC  be  268  ft.,  and  the 
angle  of  depression  at  its  top  DAC  be  33°  45'.     Required 
the  horizontal  distance  BC  and  the  vertical  height  AB. 
To  measure  an  inaccessible  height  AB  on  level  ground. 
Measure  any  distance  CD  in  a  straight  ^ 

line  towards  the  object  and  at  C  and  D  /fi 

read  the  angles  of  elevation ;    their  differ-  /'  /  \ 

ence  is  the  angle  CAD.  /'     / 

/'         / 

sin  C  X  sin  D  X  CD  c f B 

sin  (C  -  D)  FlG-  171- 

To  measure  an  inaccessible  height  which  has  no  level 
ground  before  it. 

Take  two  stations  C  and  D,  in  a  vertical  plane,  and  meas- 
ure CD]  at  C  read  the  vertical  angle  GCD  and  the  two 

vertical  angles  ACF  and  BCF. 
At  D  take  the  angle  ADE. 

Since  the  angle  EDC  =  DCG 
.'.  ADC  =  ADE  +  DCG  and 
DAC  =  ACF  -  ADE.  Then 
in  the  triangle  ADE,  the  two 
angles  ADC  and  DAC  and  the 
side  CD  are  given  to  find  the 
FlG  I?2  side  AC.  In  the  triangle  ACB 

are    given    the    angles.  ACB  = 

ACF  ±  BCF,  and  ABC  =  90°  d=  BCF,  and  the  side  AC 
to  find  the  side  AB. 

If  DE  is  above  A,  the  angle  DAC  is  the  sum  of  ACF 
and  ADE;  otherwise  it  is  their  difference.  Also  in  this 
case  ADC  is  the  difference  of  DCG  and  ADE;  otherwise 
it  is  their  sum.  Also  when  F  is  below  B,  the  angle  ACB 
is  the  difference  of  ACF  and  BCF;  otherwise  it  is  their 
sum. 

If  the  stations  C  and  D  cannot  be  conveniently  taken  in 
a  vertical  plane,  they  may  be  taken  anywhere,  and  then 
the  angles  ADC  and  A  CD  must  be  measured.  The  triangle 
A  CD  will  give  the  side  AC. 

To  measure  a  line  across  a  river  or  canyon. 
With  the  instrument  at  B,  Fig.  173,  sight  to  the  opposite 
bank  and  set  the  point  C,  so  the  points  A ,  B  and  C  will  be 


172 


PRACTICAL  SURVEYING 


in  the  same  straight  line.  An  assistant  standing  on  C 
sights  back  to  B  and  marks  out  the  line  CD  normal  to  line 
ABC.  At  D,  just  10  ft.  from  C,  set  a  point.  The  instru- 
ment man  at  B  reads  the  angle 
CBD-  then 

Length  BC  =  CD  X  cot  CBD. 

Another  method.  —  The  line  CD 
may  be  any  length  and  it  is  not 
FIG.  173.  necessary  that  the  triangle  be  right 

angled  at  C.      The  instrument  is 

set  at  B,  then  at  C  and  finally  at  D.  Each  angle  is  read. 
The  surveyor  then  has  a  triangle  with  one  side  CD,  and 
the  three  angles  known,  the  distance  from  B  to  C  being 
obtained  according  to  the  Law  of  Sines. 

To  obtain  the  depth  and  front  of  a  lot  on  a  diagonal 
street. 

The  streets  shown  in  Fig.  174  have 
an  angle  x  at  the  intersection  of  the 
lot  lines,  all  the  lines  between  the 
lots  being  normal  to  B  Avenue. 

To  obtain  the  lengths  of  the  lines  pIG   I7 

between  the  lots  from  A  Street  to  B 

Avenue  multiply  the  tangent  of  angle  x  by  the  distance  of 
the  line  from  the  point ;  for  example : 

Length  of  line  between  I  and  2  =  75  tan  xt 
Length  of  line  between  2  and  3  =  100  tan  x, 
Length  of  line  between  3  and  4  =  125  tan  x. 

To  obtain  the  frontage  on  A  Street  multiply  the  secant 
of  angle  x  by  the  width  of  the  lot  on  B  Avenue ;  for  example : 

Front  of  lot  I  on  A  Street  =  75  sec  x, 
Front  of  lot  2  on  A  Street  =  25  sec  x, 
Front  of  lot  3  on  A  Street  =  25  sec  x,  etc. 


B    Ave. 


TRAVERSES 


In  Fig.  175  it  is  desired  to  obtain  the  bearing  and  length 
of  a  straight  line  from  A  to  B.  Houses  and  shrubbery  are 
in  the  way.  The  obvious  thing  to  do  is  to  run  lines  through 


TRIGONOMETRY 


173 


the  cleared  spaces  and  calculate  the  bearing  and  length  of 
the  line  AB. 

Starting  from  Sta.  o  (A)  the  line  was  run  down  the  street 


FIG.  175.    Running  a  traverse  line. 

6*7°  oo'  E  249  ft.;  thence  N  43°  15'  E  156  ft.;  thence  to 
B,  N  11°  30'  143  ft. 

To  compute  the  missing  line  set  the  work  down  as  fol- 
lows: 


Station. 

Bearing. 

Distance. 

N+ 

S- 

E+ 

W- 

O 

I 
2 

S     7°oo'E 
N43°  is'E 
Nn°  oo'E 

249 
156 
143 

'113-63 
140.37 

247.16 

30-35 
106  .  89 
27.28 

.... 

254.00 
247.16 

247.16 

164.52 

6.84 

The  southings  lack  6.84  ft.  of  balancing  the  northings 
and  the  westings  lack  164.52  ft.  of  balancing  the  eastings. 
This  gives  a  triangle  of  the  following  shape  and  the  required 
bearing  will  be  southwest. 


174  PRACTICAL  SURVEYING 

This  is  to  be  solved  as  a  right  triangle  with  a  and  b  given, 
to  find  A ,  B  and  c ; 

C  =  90°. 

(Yesf  /64.K . 

T        „        3          6.84 

LanA  =  T  =  ~r~      —  0.0440. 
b       164.52 

.-.     A  =2°  31' 
FIG.  176.  £  =  90°  _  ^4  =  gy°  29/ 

The  required  bearing  is  S  87°  29'  W. 

ThelengthC  =  ^  =  (^=I55.77ft.         -       , 

If  it  is  necessary  to  run  the  line  from  A  to  B  without 
wanton  cutting  of  shrubbery  the  bearing  of  the  line  is 
found  in  the  manner  just  described  and  it  is  then  run  in 
from  B  towards  A ,  or  the  bearing  is  made  to  read  N  87° 
29'  E  and  the  line  is  run  from  A  to  B. 

If  the  work  is  carefully  done  the  bearing  and  distance 
should  check  with  the  computations. 

A  surveyor  must  never  neglect  to  check  his  work.  An 
experienced  surveyor  would  make  a  • "  closed  traverse ' '  by 
endeavoring  to  run  his  line  through  open  spaces  from  B 
to  4,  for  example;  then  to  5  and  back  to  A.  He  would 
compute  the  latitudes  and  departures  and  get  them  to 
balance,  as  described  in  the  chapter  on  Compass  Surveying. 
Then  the  bearing  and  length  of  line  AB  should  be  calculated 
as  already  described,  using  courses  o,  1,2,  to  Sta.  B.  An 
independent  calculation  would  then  be  made  using  courses 
3,  4,  5  to  Sta.  A.  The  differences  will  be  small  and  can  be 
averaged,  after  which  the  line  can  be  staked  out. 

"Running  a  traverse"  is  a  common  operation  in  survey- 
ing. Any  number  of  courses  may  be  run  and  the  bearing 
and  length  of  a  straight  line  joining  the  ends  of  the  first  and 
last  obtained.  In  fact  the  bearing  and  length  of  a  line  join- 
ing any  two  points  on  a  traverse  may  be  .found  by  tabulat- 
ing only  the  courses  involved. 

The  captain  of  a  ship  finds  her  position  by  traversing, 
the  work  of  surveyors  and  navigators  being  in  many 
respects  similar. 


TRIGONOMETRY 


175 


OMISSIONS 

The  supplying  of  omitted  bearings  and  distances  in  sur- 
veys is  an  extension  of  the  principles  of  traversing. 

Many  reasons  may  be  given  to  explain  why  a  portion  of 
the  field  notes  are  omitted  but  a  principal  reason  is  forget- 
fulness,  which  is  one  manifestation  of  carelessness.  In  the 
example  under  the  head  of  traversing  one  reason  for  the 
omission  to  run  one  course  was  given.  It  sometimes  hap- 
pens that  two  courses  may  lie  in  swampy,  or  otherwise 
inaccessible,  places,  in  which  case  the  surveyors  should  run 
a  closing  line  to  make  a  closed  survey  of  all  accessible 
corners,  the  inaccessible  courses  then  forming  with  the 
closing  line  a  separate  survey.  All  errors  are  thrown  into 
the  omitted  parts. 

There  are  four  cases  in  omissions,  each  of  which  will  now 
be  illustrated. 

CASE  I .  —  Bearing  and  length  of  one  course  omitted. 


Station. 

Bearing. 

Distance. 

Latitudes. 

Departures. 

N+ 

S- 

E+ 

W- 

0 
I 
2 

3 
4 

Ni6°  30'  E 
N82°  05'  E 
S  16°  54'  E 
S  36°  58'  W 
Omitted 

22.  IO 
19.62 

23-97 
22.  II 

Omitted 

21.190 
2.702 

6.276 

19-433 
6.968 

22-935 
17.666 

13.296 

23.892 

40.601 
23-892 

32.677 
13.296 

13.296 

16.709 

I9-38I 

Nat.  tan  of  bearing  = 


Length  of  course 


Bearing  =  TV  49°  14'  W. 
dep. 


1.15991 


19.381 


sin  bearing      0.75738 


=  25.59  chains. 


The  traverse  table  in  the  chapter  on  compass  surveying 
gives  values  to  quarter  degrees  only,  whereas  in  work  done 
with  a  transit  angles  are  read  to  one  minute  of  arc,  except 
in  extremely  high-grade  work  when  angles  are  read  to  one 


176  PRACTICAL  SURVEYING 

second.  Surveyors'  transits  usually  read  to  minutes  and 
engineers'  transits  read  to  one-half  or  one-third  of  a  minute. 
When  traversing  is  done  by  means  of  an  instrument  reading 
closer  than  a  compass  it  is  plain  that  a  compass  traverse 
table  cannot  be  used. 

It  is  common  to  use  a  table  of  sines  and  cosines,  for 

Cosine  X  distance  —  latitude. 
Sine  X  distance  =  departure. 

The  student  must  memorize  this. 

In  all  tables  of  latitudes  and  departures,  for  any  angle, 
the  value  given  in  column  I  for  the  latitude  is  the  cosine  of 
that  angle,  and  the  value  given  in  column  I  for  the  depar- 
ture is  the  sine  of  that  angle. 

The  actual  labor  of  multiplying  and  dividing  can  be 
greatly  lessened  by  using  Crelle's  Tables  ($5.00),  by  using 
a  slide  rule,  or  by  logarithms.  Crelle's  Tables  contain  the 
products  of  all  numbers  up  to  1000  X  1000  and  by  following 
the  instructions,  products  of  much  larger  numbers  may  be 
obtained  by  inspection.  The  tables  are  as  readily  used  for 
division  of  one  number  by  another.  The  slide  rule  is  not 
so  good  for  this  class  of  problems  as  for  others  and  is  not 
recommended  for  traversing  calculations  except  for  check- 
ing, in  which  work  it  is  better  than  a  trigonometer.  Log- 
arithms are  great  labor  savers  and  the  degree  of  accuracy 
of  logarithmic  work  is  fixed  by  the  number  of  significant 
figures  to  which  the  tables  are  computed. 

The  greatest  labor  savers  in  calculating  latitudes  and 
departures  are  traverse  tables,  which  reduce  all  multiplica- 
tion to  addition.  The  author  has  read  in  several  textbooks 
written  by  teachers,  that  traverse  tables  do  not  save  time 
as  compared  with  tables  of  logarithms  and  logarithmic 
functions  of  angles.  For  a  number  of  years  the  work  of 
the  author  was  principally  surveying  and  he  found  that 
traverse  tables  are  far  superior  to  logarithms  in  saving 
time  and  lessening  liability  of  errors. 

The  oldest  known  traverse  tables  with  Latitude  and  De- 
parture computed  for  each  minute  of  angle  are  those  of 
Gen.  J.  T.  Boileau  ($5.00).  In  1900  H.  Louis  and  G.  W. 
Caunt  brought  out  a  book  somewhat  more  convenient  to 


TRIGONOMETRY 


177 


use,  with  larger  and  clearer  type.  The  price  of  this  book 
is  two  dollars,  the  page  measuring  six  by  nine  inches. 

The  most  complete  Traverse  Tables  are  those  of  R.  L. 
Gurden.  The  latitudes  and  departures  are  computed  to 
four  places  of  decimals  and  for  every  minute  of  angle  up  to 
100  of  distance.  The  size  is  9  ins.  by  14  ins. ;  the  binding  is 
cloth  and  half  morocco.  When  the  author  bought  his  copy 
in  1889,  the  price  was  $12.50,  but  is  now  $7.50.  The  Louis 
and  Caunt  tables  are  admirable  for  field  use  while  the  Gur- 
den tables  are  ideal  for  the  orifice.  The  difference  in  the 
tables  lies  in  the  fact  that  the  Boileau  and  the  Louis  and 
Caunt  tables  are  computed  only  for  all  distances  up  to 
10.  The  following  example  will  serve  as  an  illustration. 

Find  the  latitude  and  departure  of  S  1 6°  54'  E  23.97 
chains. 


3.   Gurden. 
16°  54' 


20.00 
3.00 
0.90 
0.07 


i.    Boileau. 
2.    Louis  &  Caunt. 
16°  54' 

Lat. 

Dep. 

i         I9-I36 

>            2  .  8704 

5.814 
0.8721 

i            0.86II3 
0.06698 

0.26163 
0.02035 

22.93451 

6.96808 

Lat.  Dep. 

23.00        22.0064  6.6861 

0.97          0.92811  0.28198 

22.93451  6.96808 


To  obtain  the  same  results  by  using  logarithms  it  will  be 
necessary  to  take  from  one  table  the  logarithm  of  23.97 
and  from  another  table  the  logarithm  of  the  sine  and  of 
the  cosine  of  16°  54'.  These  are  added  and  the  numbers 
corresponding  to  the  resulting  logarithms  found. 

By  logarithms: 

Dep. 

=  1.379668 
=  9.463448 

10.843116 

10. 

=  0.843116 
=  6.968 

CASE  II.  —  Lengths  of  two  courses  omitted. 
This  might  be  a  case  where  bearings  are  taken  to  a  cor- 
ner, which  is  visible  but  inaccessible. 


log.  23.97 
log.  cos  1  6°  54' 

subtract 
log.  lat. 
Latitude 

Lat. 
-    1.379668 
=    9.980827 
11.360495 

10. 

log.  23.97 
log.  sin  16°  54' 

subtract 
log.  dep. 
Departure 

=    1.360495 
=  22.9345 

178 


PRACTICAL  SURVEYING 


In  the  assumed  problem  the  two  courses  are  adjacent. 
When  the  courses  are  not  adjacent  the  method  is  the  same, 
or  the  method  for  Case  IV  may  be  used. 

If  the  two  sides  are  parallel  the  problem  is  indeter- 
minate. 

The  whole  process  in  Cases  II,  III  and  IV  is  to  discard 
the  omitted  parts  and  calculate  the  bearing  and  length  of 
a  closing  line.  A  triangle  is  then  formed  on  this  closing 
line  as  a  base  and  the  missing  parts  computed. 


Station. 

Bearing. 

Distance. 

Latitudes. 

Departure. 

N+ 

s- 

E+ 

W- 

O 

I 
2 

3 

4 

Ni6°  30'  E 
N82°  05'  E 
S  16°  54'  E 
S  36°  58'  W 
N49°  14'  W 

22.  IO 

19.62 

23-97 

Omitted 
Omitted 

21  .190 
2.7O2 

6.276 

19-433 
6.968 

22.935 



23.892 
22-935 

22.935 

32.677 

0-957 

Nat.  tan  of  bearing  = 


Length  of  course 


Bearing  =  S  88°  19'  W. 
dep. 


34-14524- 


32.677 


sin  bearing      sin  88°  19' 

=  3  '   ''    =  32.691  chains. 
0-99957 


B/ 


FIG.  177- 


Draw  a  triangle  (free  hand)  as  shown 
in  Fig.  177  and  from  the  bearings  com- 
pute angles  A  ,  B  and  C. 

S88°i9'W   =  S87°79'W 
S  36°  58'  W 

angle  B. 


51°  21' 


85°  72'      =  86°  12'  =  angle  C 


TRIGONOMETRY 


I79 


S  88° 
N49° 


19'  W 
14'  W 


To  Check 
°         ' 


137°  33' 
°6o' 


12 


179 


42°  27' 


& 

42° 

=  angle  A  179° 

sin  A       sin  5 


32.691 


0     ,/  32.691    \     0.67495X32.691 

.'.  a  =  sin  42°  27f(-^-—~ — 7  =— —  —  =  22. 1 1  chains 

'  \sin86  127  0.99780 


and 


.78098X32.691 

099780    — 


CASE  III.  —  Bearings  of  two  courses  omitted. 

This  problem  may  have  two  solutions  but  the  ambiguity 
is  practically  unimportant  as  the  surveyor  is  presumed  to 
have  a  clear  idea  of  the  shape  of  the  field. 


Station. 

Bearing. 

Distance. 

Latitudes. 

Departures. 

N+ 

s- 

E+ 

W- 

O 
I 
2 
3 
4 

Ni6°  30'  E 
N82°  05'  E 
S  16°  54'  E 
Omitted 
Omitted 

22.  TO 

19.62 

23-97 
22.11 

25-59 

21  .190 
2.7O2 

6.276 

19-433 
6.968 

22-935 

23.892 
22-935 

32.677 

0-957 

This  gives  the  same  bearing  and 
length  for  the  closing  line  as  were 
found  for  the  problem  illustrating 
Case  II.  The  triangle  is  shown  in 
Fig.  178.^ 

If  the  instrument  man  does  not  make 
a  sketch  in  his  notes  to  guide  the  com- 
puter in  the  office,  the  latter  might 
assume  the  courses  to  run  as  indicated 
by  the  dotted  lines.  The  interior  angles 


-4V 


FIG.  178. 


i8o 


PRACTICAL  SURVEYING 


of  the  closing  triangle  will  not  be  affected  but  the  shape  and 
area  of  the  field  will  differ  considerably  from  the  truth. 


Lei  s  = 


2559  +  32.69 


A  =  42°  28'. 


I4.6i  X  7-5i 
J5-59  X  32.69 


=  40  20 


=  0.36216. 


fiSM 

V  22.11 


09  X  7-5i 


ac  r  22.11  X  32.69 

B  =  51°  22'. 

C  =  180°  -(A+B)  =  179°  60'  -  93°  50' 


=  0.43355. 


86°io/. 


S  88°  19'  W 

51°    22'  L 


a  =  S  36°  57: 


86°  10'  L 

836°  47' W  =  N  36°  57'  E 
b  =  N49°  13' W 


CASE  IV.  —  Length  of  one  course  and  bearing  of  another 
omitted. 

The  field  is  to  be  revolved  until  the  course  of  which  only 
the  bearing  is  known  is  in  the  meridian,  thus  throwing  all 
the  unknown  departure  into  the  course  of  which  only  the 
length  is  known.  The  omitted  parts  may  be  on  adjacent  or 
non-adjacent  courses. 


Station. 

Bearing. 

Revolved  to 
left. 

New  bearing. 

Distance. 

0 

Ni6°  30'  E 

36°  58' 

N  20°   28'  W 

22.  IO 

I 

N82°  05'  E 

36°  58' 

N45°  07'  E 

19.62 

2 

S  16°  54'  E 

36°  58' 

S  53°  52'  E 

23-97 

3 

4 

S  36°  58'  W 
Omitted 

36°  58' 

South 
Omitted 

Omitted 

2$  •  "?Q 

When  the  computations  are  finished  a  doubt  sometimes 
arises  as  to  whether  the  latitude  of  the  course  whose  bear- 
ing is  omitted  is  a  northing  or  a  southing.  This  produces 
two  sets  of  values,  either  of  which  will  satisfy  the  problem, 
though  one  will  give  a  wrong  area.  The  knowledge  the 
surveyor  has  of  the  general  directions  of  the  courses  and 
the  shape  of  the  field  must  settle  all  such  ambiguities. 


TRIGONOMETRY 


181 


The  habit  of  making  free-hand  sketches  to  supplement  the 
field  notes  is  one  every  instrument  man  should  early  ac- 
quire. 


Sta. 

Bearing. 

Distance. 

Latitudes. 

Departures. 

N+ 

S- 

E+ 

W- 

o 

I 

2 

3 

4 

N  20°   28'  W 

N4S°o7/E 
S  53°  52'  E 
South 
Omitted 

22.10 
19.62 
23-97 

Omitted 
25-59 

20.705 
I3-845 

H.I34 

13.902 
19-360 

7.727 

34-550 
14.134 

14-134 

33.262 

7.727 

7.727 

20.416 

25-535 

All  the  difference  in  departure  is  thrown  into  course  4, 
the  length  of  which  is  25.59  chains.  The  following  triangle 
is  obtained. 

The  sine  of  A  =  - 


thus    making    the    bearing    of 
course  4  =  86°  14'  but  some  un-  FIG.  179. 

certainty  exists  as  to  whether  it 

is  a  northing  or  a  southing.     From  the  differences  in  lati- 
tudes and  departures  it  is  apparently  S  86°  14'  W. 
To  settle  the  matter  apply  the  angle  of  revolution. 


86° 
36° 


14' W 


123 
179 


12' 
60' 


N    56°  48'  W 

The  bearing  was  known  (from  the  study  of  Cases  I,  II 
and  III)  to  be  N  49°  14'  W  so  another  trial  must  be  made. 


N86° 
36° 


14'  W 

58' R 


N  49°  16'  W 


182 


PRACTICAL  SURVEYING 


The  bearing  is  evidently  a  northing.  The  difference  of 
two  minutes  is  due  to  the  computations  in  some  of  the 
examples  being  carried  to  a  greater  degree  of  refinement 
than  in  others.  It  will  make  no  appreciable  difference  in 
the  length  of  course  4,  but  the  difference  in  angle  introduces 
a  closing  error  of  about  ij  links.  No  survey  is  free  from 
error,  but  all  errors  being  thrown  into  the  courses  supplied 
by  computation  a  false  idea  of  accuracy  is  obtained  by 
doing  work  in  the  office  that  belongs  to  the  field.  "Meas- 
ure in  haste  and  repent  in  the  office, "  is  a  true  proverb. 

The  latitude  of  the  course  =  cosine  86°  16'  X  25.59  = 
1 .68 1,  and,  this  amount  is  to  be  added  to  the  latitude 
difference  to  obtain  the  southing  for  course  3,  so  the  north- 
ings and  southings  will  balance. 

Latitude  for  course  3  =  20.416  +  1.681  =  22.097. 

The  latitudes  and  departures  are  now 


Station. 

Lati.uJes. 

Departures. 

+ 

- 

+ 

- 

0 

I 
2 

3 
4 

20.705 
I3-84S 

7.727 

14-134 
22.097 

13.902 
19.360 

25-535 

I.68I 

36.231 

36.231 

33.262 

33.262 

and  the  bearings  may  be  restored  by  revolving  the  field 
to  the  right  through  an  angle  of  36°  58'. 

The  field  is  not  altered  in  size  or  shape  by  being  revolved 
so  the  area  may  be  computed  by  using  the  values  already 
found  for  the  latitudes  and  departures. 

LOGARITHMS 

A  logarithm  is  an  exponent,  and  in  an  earlier  chapter  it 
was  shown  that 


and 


a. 


TRIGONOMETRY  183 

A  series  of  quantities  which  increase  or  decrease  by  a 
common  difference  is  called  an  Arithmetical  Progression-; 
as  i,  2,  3,  4,  5,  etc.,  or  75,  72,  69,  66,  etc. 

A  series  of  quantities  which  increase  by  a  constant  multi- 
plier, or  decrease  by  a  constant  divisor,  is  called  a  Geometri- 
cal Progression;  as  2,  8,  32,  128,  etc.,  the  multiplier  being 
4;  or  567,  189,  63,  etc.,  the  divisor  being  3. 

We  can  write  down  a  line  of  figures  in  arithmetical  pro- 
gression over  a  line  in  geometrical  progression,  and  the 
upper  line  will  contain  the  exponents  of  the  second  line  so 
that  multiplication  of  two  quantities  in  the  lower  line  may 
be  accomplished  by  adding  their  exponents. 

0.5  0.4  0.3         02        o.i      01234  5  6 

O.OOOOI       O.OOOI       O.OOI       O.OI       O.I        I        IO       IOO       IOOO       IO.OOO        IOO.OOO       I.OOO.OOO 

Add  2  and  3,  the  sum  is  5.  Under  5  is  found  100,000 
which  is  plainly  the  product  of  100  (under  2)  and  1000 
(under  3). 

Subtract  4  from  5,  the  difference  is  i.  Under  5  is  found 
100,000  and  under  4  is  found  10,000.  Under  I  is  found 

100,000 
10  =  -       — • 
10,000 

Multiply  2  by  3,  the  product  is  6.  Under  6  is  found 
1,000,000  which  is  the  square  of  1000,  found  under  3. 

Divide  6  by  2,  the  quotient  is  3.  Under  3  is  found  1000 
which  is  the  square  root  of  1,000,000  found  under  6. 

The  student  for  an  exercise  may  carry  these  series,  as- 
cending and  descending,  to  10  or  more  places  each  way, 
and  multiply  and  divide,  raise  to  powers  and  extract  roots, 
keeping  always  in  an  ascending  or  a  descending  series. 

Logarithms  are  a  series  of  artificial  numbers  used  as 
exponents  for  numbers  placed  in  arithmetical  progression. 
By  their  use  addition  takes  the  place  of  multiplication  and 
subtraction  takes  the  place  of  division.  Numbers  are 
raised  to  any  power  by  multiplying  the  logarithm  of  the 
number  by  the  exponent  representing  that  power,  and  roots 
are  extracted  by  dividing  the  logarithm  by  the  index  of 
the  required  root. 

Logarithms  are  useful  for  many  calculations,  especially 
those  involving  proportion,  or  a  combination  of  multi- 
plication and  division. 


184  PRACTICAL  SURVEYING 

An  expression  containing  a  positive  or  negative  sign  is  not 
in  shape  for  logarithmic  computation  and  must  be  differ- 
ently written  to  eliminate  every  such  sign.  For  example 

sin  (x  +  y)  +  sin  (x  —  y) 

can  be  adapted  for  logarithmic  computation  by  writing  it 
as  follows : 

2  sin  x  X  cos  y. 

Adding  logarithms  is  equivalent  to  multiplying  their 
numbers,  and  subtracting  logarithms  is  equivalent  to  per- 
forming the  operation  of  division  with  their  numbers. 

Some  surveyors  never  use  logarithms.  Other  surveyors 
use  them  upon  every  occasion.  Whether  to  use  "logs"  or 
"naturals"  is  something  each  man  must  settle  for  himself 
and  is  not  worth  discussing.  For  involved  calculations 
logarithms  save  time  and  lessen  the  liability  of  error. 

TO  USE  A  TABLE  OF  LOGARITHMS 

A  complete  logarithm  consists  of  two  parts.  The  man- 
tissa is  the  decimal  part  printed  in  the  table.  The  charac- 
teristic is  a  whole  number  and  depends  upon  the  number 
of  figures  in  the  number. 

The  log.  of  2500  is  3.3979  of  which  0.3979  is  the  mantissa 
taken  from  the  table  and  3  is  the  characteristic. 

The  log.  of  8435  =  3.926085.  The  log.  of  8.435  =  0.926085. 
The  log.  of  843.5  =  2.926085.  The  log.  of  0.8435  =  —  1.926085. 
The  log.  of  84.35  =  1.926085.  The  log.  of  0.08435  =  -  2.926085. 

The  characteristic  is  seen  by  the  above  illustration  to  be 
a  number  indicating  I  less  than  the  number  of  integral 
figures  of  which  the  number  consists.  In  decimal  num- 
bers the  characteristic  is  negative  and  indicates  the  distance 
of  the  first  significant  figure  from  the  decimal  point. 

In  some  tables  of  logarithms  there  is  a  column  headed 
"Tabular  Difference."  In  the  table  in  this  book  the 
tabular  difference  is  not  shown.  The  logarithm  of  243  = 
2.38561  and  the  logarithm  of  244  =  2.38739.  The  differ- 
ence is  0.00178  and  would  be  printed  as  Tab.  Diff.  178. 
The  Tabular  Difference  for  each  horizontal  line  is  therefore 
the  average  difference  between  the  units  figures  on  the  line. 


TRIGONOMETRY  185 

TO  FIND   THE  LOGARITHM   OF  A  NUMBER 

First  fix  the  characteristic.  Then  look  in  the  table  for 
the  mantissa.  The  number  may  have  only  three  figures, 
in  which  case  the  mantissa  will  be  found  at  once.  If  it 
contains  four  or  more  figures  the  mantissa  for  the  three 
first  figures  is  found  and  the  difference  between  this  and  the 
next  higher  figure  used  in  the  following  way : 

Find  the  log.  of  2369. 

The  characteristic  =  3.  The  mantissa  for  2360  = 
0.37291.  To  find  the  mantissa  find  23  in  the  column 
headed  "No.,"  and  proceed  horizontally  to  the  right  to 
the  column  headed  "6." 

The  remainder  of  9  cannot  be  ignored.  Find  the  man- 
tissa for  237  =  0.37475.  The  difference  between  0.37475 
and  0.37291  =  0.00184  and  this  must  be  multiplied  by  0.9  = 
0.001656,  the  product  being  added  to  the  mantissa  for  2360. 
The  complete  log.  is  as  follows : 

Mantissa  of  236  =  0.37291 
Mantissa  of  0.9    =  0.00166 

0-37457 
Log.  of  2369         =  3-37457- 

The  reason  for  designating  the  9  remainder  as  0.9  may  be 
found  by  considering  the  mantissas  as  being  given  in  the 
table  for  2370  and  for  2360,  but  not  for  fractional  parts, 
therefore  9  is  nine-tenths  the  difference  between  60  and  70. 
Had  the  number  been  236,924  the  process  would  have  been 
the  same  but  the  characteristic  would  be  5  and  the  differ- 
ence 0.00184  would  be  multiplied  by  0.924  =  0.00170. 

Mantissa  of  236     =  0.37291 
Mantissa  of  0.924  =  0.00179 

0.37470 
Log.  of  236924       =  5-37470. 

Tables  of  logarithms  having  a  column  of  tabular  differ- 
ences show  the  difference  of  0.00184  as  a  whole  number, 
184,  to  save  space. 

The  student  must  remember  that  the  mantissa  is  a  deci- 
mal fraction  and  therefore  the  tabular  difference  is  a  decimal 
fraction  containing  the  same  number  of  figures,  the  differ- 


1  86  PRACTICAL  SURVEYING 

ence  being  made  up  by  placing  ciphers  in  front  of  the 
significant  figures. 

To  find  the  number  corresponding  to  a  given  logarithm. 

The  log.  is  3-37457- 

In  the  table  of  logs,  find  the  mantissa  nearest  to  the  given 
mantissa  but  below.  This  we  find  to  be  0.37291,  corre- 
sponding to  number  2360,  for  as  we  have  3  for  a  character- 
istic there  must  be  four  figures  in  the  number. 

Given  log.  =  3-37457 

Log.  of  2360       =  3.37291 
Diff.  =  0.00166 

Tabular  difference  between  mantissas  of  2370  and  2360  = 
0.00184. 

0.00166 


which  gives  a  number  =  2369.13.  The  0.13  is  an  exceed- 
ingly small  per  cent  of  2369  but  proves  that  a  table  of 
logarithms  of  numbers  from  o  to  1000  is  not  accurate  for 
numbers  containing  more  than  four  figures,  when  the  man- 
tissa is  carried  to  only  five  places.  Tables  of  logarithms 
from  o  to  10,000  are  accurate  for  numbers  containing  as 
many  figures  as  there  are  decimal  places  in  the  mantissa. 
Tables  of  logarithms  from  o  to  108,000  are  accurate  enough 
for  practical  use  for  numbers  containing  one  figure  more 
than  the  decimal  figures  in  the  mantissa.  For  all  the  work 
of  surveyors  the  five-place  table  here  given  is  accurate 
enough  for  all  practical  purposes  for  numbers  containing 
not  more  than  five  figures.  A  table  for  numbers  from  o 
to  10,000  is  more  convenient  to  use.  In  the  examples 
following  notice  the  inaccuracy  of  the  work  due  to  the 
table  used. 

To  multiply  two  numbers  by  using  logs. 

I.   54.3  X  6.19  =     ? 

Log.  54.3         =  1.73480 

Log.    6.19       =  0.79169 

2.52649 

Log.  of  336      =  2.52633 
0.00016 


TRIGONOMETRY  187 

Tab.  diff.  336  and  337  =  0.52763  -  0.52633  =  0.00130. 

0.00016 

-  =  0.123. 
0.00130 

By  logs.  54.3  X  6.19  =  336.123. 
By  arithmetic  =336.117. 

2.  54.3  X  6.19  X  27  =  ? 

Log.  54.3  =  1.73480 

Log.  6.19  =  0.79169 

Log.  27  =  1.43136 

3.95785 
Log.  of  9070  =  3.95761 

0.00024 

Tab.  diff.  9070  and  9080  =  0.95809  —  0.95761  =  0.00048. 
0.00024  _ 
0.00048  ~ 

By  logs.  54.3  X  6.19  X  27  =  9075.000. 
By  arithmetic  =  9075.159. 

The  rule  for  multiplying  numbers  by  using  logs,  is  to  add 
the  logarithms  of  the  numbers  and  find  from  the  table  the 
number  corresponding  to  the  new  logarithm  thus  obtained. 

To  divide  a  number  by  another  by  using  logs. 

54-3  =  ? 

6.19 

Log.  54.3         =  i.7348o 
Log.  6.19         =  0.79169 

0.943H 
Log.  8.77         =  0.94300 

Diff.  =  o.oooi  i 

Tab.  diff.  878  and  877  =  0.94349  —  0.94300  =  0.00049. 

o.oooi  i 

-  =  0.225. 
0.00049 

By  logs.  54.3  +  6.19  =  8.77225. 
By  arithmetic  =  8.77205. 

To  divide  one  number  by  another  find  the  difference 
between  their  logarithms.  The  resulting  logarithm  is  the 
log.  of  the  quotient  of  the  numbers. 


l88  PRACTICAL  SURVEYING 

To  raise  a  number  to  any  power. 
Find  the  square  of  6.19. 

Log.  6.19     =  0.79169 

2 

1.58338 

Log-  38.3     =  1-58320 
0.00018 

Diff.       =  0.00018  _ 
Tab.  diff.  "  0.00133  " 

6TQ2(By  logs.  =  38.3135- 

9  (By  arithmetic         =  38.3161. 

The  rule,  which  is  general,  is  to  multiply  the  log.  by  the 
exponent  of  the  power.  The  number  corresponding  to  the 
log.  thus  obtained  is  the  power  sought. 

To  extract  the  root  of  any  number. 

What  is  the  square  root  of  54.3? 

Log.  54-3  =  '-^  =  0.86740 
Log.  7.36  =  0.86688 

0.00052 

=  0-QOQ52  =  Q  8gl 
Tab.  diff.      0.00059 
/—-(By  logs.  =7.36881. 

v543JBy  arithmetic       =  7.368  +  . 

The  rule,  which  is  general,  is  to  divide  the  log.  of  the 
number  by  the  index  of  the  root.  The  number  correspond- 
ing to  the  log.  thus  obtained  is  the  root  sought. 

ARITHMETICAL   COMPLEMENT 

A  logarithm  may  be  mentally  subtracted  from  10,  an 
integer,  or  the  right-hand  figure  may  be  subtracted  from 
10,  and  all  the  rest  from  9. 

By  using  the  arithmetical  complement  of  a  logarithm, 
division  instead  of  being  subtraction  becomes  addition. 
That  is,  adding  an  arithmetical  complement  is  equivalent 
to  subtracting  its  logarithm.  In  the  product,  10  must  be 
subtracted  from  the  characteristic. 


TRIGONOMETRY  189 

NEGATIVE  CHARACTERISTIC 

The  characteristic  is  negative  when  the  number  is  a 
decimal  fraction. 

A  negative  characteristic  must  be  subtracted  when  the 
logarithm  is  added  and  must  be  added  when  the  logarithm 
is  subtracted. 

After  multiplying  a  negative  index  subtract  from  the 
resulting  index  the  amount  carried  from  the  mantissa. 

Example.  —  Raise  0.009  to  the  third  power. 

Log.  0.009  =  3-95424 

3 

7.86272 

The  —  7  was  obtained  as  follows :  2  was  carried  from  the 
multiplication  of  the  mantissa.  The  product  of  3  X  (—3) 
=  —  9  and  +2  —  9=— 7.  It  is  customary  to  place  the 
negative  sign  above  the  characteristic  instead  of  placing 
it  in  front. 

Sometimes  a  positive  characteristic  is  used,  in  which 
case  10  times  the  exponent  of  the  power  lessened  by  I 
must  be  taken  from  the  final  characteristic,  and  the  result 
added  to  — 10. 

Example.  —  Raise  0.0437  to  the  fourth  power. 

Neg.  char.  Pos.  char. 

Log.  0.0437      =  2.64048  or    8.64048 

= 4     4 

Log.  0.000003649  =  6.56192     +  4.56192 

—  IP 

6.56192 

The  6  was  obtained  when  the  negative  characteristic  was 
used  by  subtracting  the  2  carried  from  the  mantissa  from 
the  product  of  —2  X  4  =  —8. 

In  the  case  of  the  positive  characteristic  4  X  8  =  32  and 
adding  the  2  carried  from  the  mantissa  the  result  was  34. 
From  this  was  subtracted  10  X  (4  —  i)  =  30.  Then 
+4  —  10  =  —6. 

When  extracting  roots  if  the  given  number  be  a  decimal, 
and  its  characteristic  positive,  subtract  I  from  the  index  of 


190  PRACTICAL  SURVEYING 

the  root  and  add  the  remainder  to  the  characteristic  before 
dividing. 

If  the  characteristic  be  negative,  add  to  it  the  least  num- 
ber that  will  make  the  sum  divisible  by  the  index  of  the 
root;  the  quotient  is  the  characteristic  of  the  log.  of  the 
root.  In  dividing  the  mantissa,  only  the  number  added  is 
to  be  considered  as  the  characteristic. 


Example.—  ^0.00130321  =  ? 
Log.  0.00130321  =  3 
Add  i 


Divide  by  4  =  1.27875 

The  method  by  which  the  characteristic  I  was  obtained 
is  clearly  seen.  In  dividing  the  mantissa  the  character- 
istic 3  was  ignored  and  the  I  substituted. 

To  WORK  PROPORTION  BY  LOGARITHMS 

Add  the  logarithms  of  the  second  and  third  terms  to- 
gether. From  their  sum  subtract  the  logarithm  of  the  first 
term.  The  remainder  is  the  logarithm  of  the  fourth  term. 

Or;  add  together  the  arithmetical  complement  of  the 
first  term  and  the  logarithms  of  the  other  two.  The  sum, 
with  10  subtracted  from  the  characteristic,  is  the  logarithm 
of  the  answer. 

Example.  —  Illustrating  the  two  methods. 

36   :    144   ::  28   :  x 
Log.  144    =  2.15836  2.15836 

Log.    28    =  1.44716  1.44716 

3-60552 
Log.    36    =  1.55630  a.c.  =  8.44370 

2.04922  2.04922 

X  =   112. 

(The  10  is  subtracted  mentally.) 

LOGARITHMIC  FUNCTIONS   OF  ANGLES 

Logarithmic  functions  of  angles  are  merely  logarithms 
of  the  natural  functions.  Characteristics  are  placed  in  the 
tables  but  as  the  natural  functions  are  decimal  fractions 


TRIGONOMETRY  191 

the  actual  characteristics  are  negative.  The  numbers 
used  as  characteristics  are  positive  for  convenience,  being 
the  difference  between  10  and  the  true,  negative  charac- 
teristic. Tables  of  logarithmic  functions  are  for  use  with 
tables  of  logarithms  of  numbers. 

Example.  —  Refer  to  example  in  Case  III  of  Omissions, 
and  compare  the  work  required  when  using  "naturals" 
and  using  "logs.  " 


\l 

V 


I4>61  x 


25-59  X  32-69 
Log.  14.61  =  1.16465 
Log.    7.51  =  0.87564  2.04029 

Log.  25.59  =  140807 
Log.  32.69  =  1.51442 

2.92249  a.c.  =  7.07751 

9.11780  =  1.11780 
1.11780 
i 

T.55890  =  9-55890  =  sin  21°  14' 
A  =  21°  14'  X  2  =  42°  28'. 

The  student,  for  exercise,  should  work  all  the  examples  in 
this  chapter  by  using  logarithms.  In  this  way  a  compar- 
ison can  be  made  of  the  advantages  and  disadvantages  of 
logarithms  and  their  limitations  in  some  kinds  of  work, 
bearing  in  mind  the  degree  of  accuracy  possible  with  the 
small  table  of  logs,  in  this  book. 

PROPORTIONAL  PARTS  IN  LOGARITHMIC   TABLES 

The  Five  Place  Logarithmic-Trigonometric  Tables  by 
Constantine  Smoley,  C.E.,  are  printed  on  a  thin,  tough 
bond  paper  and  are  bound  in  flexible  cloth.  The  price  is 
50  cents.  They  consist  of  logarithms  of  numbers  from 
I  to  10,000;  Logarithms  of  the  sine  and  tangent  varying  by 
ten  seconds  from  o°  to  3°  and  of  the  cosine  and  cotangent 
varying  by  ten  seconds  from  87°  to  90°;  logarithms  of  the 
sine,  cosine,  tangent  and  cotangent,  secant  and  cosecant 
for  each  minute  of  arc;  tables  of  squares,  cubes,  square  and 
cube  roots,  etc. 


I92 


PRACTICAL  SURVEYING 


In  the  table  of  logarithms  of  numbers  a  star  is  frequently 
used  as  follows: 


N. 

o 

i 

2 

3 

4 

5 

6 

7 

8 

9 

426 

941 

951 

96l 

972 

982 

992 

*002 

*OI2 

*022 

*033 

427 

63,043 

053 

063 

073 

083 

094 

104 

114 

124 

134 

The  mantissa  is  carried  to  five  figures  but  the  first  two 
figures  are  omitted  wherever  possible  in  order  to  save  space 
and  make  the  tables  more  easily  read.  The  mantissa  of 
4260  is  62,941,  of  4265  it  is  62,992,  etc.  The  star  in  front  of 
the  mantissa  for  4266  indicates  that  the  first  two  figures 
are  increased,  so  this  mantissa  is  63,002;  the  mantissa  for 
4268  is  63,022,  etc. 

On  the  right  hand  of  each  page  is  a  column  headed  P.P. 
(Proportional  Parts)  in  which  figures  appear  as  follows: 


10 


I.O 
2.O 

3-0 
4-0 

.0 
.0 

7-o 
8.0 
9.0 


i:: 


Example.  —  Find  the  log.  of  426,758. 
The  tabular  difference  between  logs,  of  4267  and  4268  = 
10,  which  number  heads  the  P.P.  column. 

Log.  of  426,700  =  5.63012 
P.P.  50  =  0.000050 

P.P.    8  =  0.000008 

5.630178 

(Note.  —  These  values  of  P.P.  for  differences  of  50  and  8 
apply  only  to  tab.  diff.  of  10. 

When  tab.  diff.  =  14  then  P.P.  for  5  =  7.0  and  for  8  = 
1 1. 2,  etc.) 


TRIGONOMETRY  193 

Example.  —  Find  number  corresponding  to  log.  5.63018. 

Log-  5-63018 

No.  426,700  =  Log.  5.63012 
0.00006 

The  difference  between  log.  of  426,700  and  426,800  =  10, 
so  in  the  P.P.  column  find  10.  On  the  right  of  the  vertical 
line  in  the  column  find  6  and  on  the  left  of  the  line  is  found 
6.0  which  is  placed  after  the  7  in  the  number  and  we  thus 
obtain 

No.  426,760  log.  =  5.63018. 

In  the  tables  of  logarithms  of  functions  of  angles  the  P.P. 
column  refers  to  seconds.  Few  surveyors  or  engineers 
read  angles  closer  than  the  nearest  minute  and  few  problems 
are  encountered  in  which  seconds  appear  in  the  angles. 
When  the  angles  do  contain  seconds  the  proportional  parts 
are  used  in  the  way  just  described  for  logarithms  of  num- 
bers. For  angles  smaller  than  3°  the  differences  are  so 
great  between  consecutive  minutes  that  a  separate  table  is 
given  in  which  the  values  of  the  functions  differ  by  10" 
and  the  P.P.  column  gives  the  differences  for  single  seconds. 
With  the  explanation  given,  the  student  can  with  a  little 
practice  use  the  columns  of  proportional  parts  readily. 


194 


PRACTICAL  SURVEYING 
NATURAL  FUNCTIONS  OF  ANGLES 


Deg. 

Min. 

Sine. 

Cosec. 

Tang. 

Cotang. 

Sec. 

Cosine. 

Min. 

Deg. 

0 

0 

o.oooooo 

Infinite. 

o.oooooo 

Infinite. 

.00000 

I.  000000 

0 

90 

10 

0.002909 

343.77516 

0.002909 

343-77371 

.00000 

0.999996 

So 

20 

0.005818 

171.88831 

0.005818 

171.88540 

.00002 

0.999983 

40 

30 

0.008727 

114.59301 

0.008727 

114.58865 

.00004 

0.999962 

30 

40 

0.011635 

85.945609 

0.011636 

85.939791 

.00007 

0-999932 

20 

So 

0.014544 

68.75736o 

0.014545 

68.750087 

.OOOII 

0.999894 

IO 

1 

o 

0.017452 

57.298688 

0.017455 

57.289962 

.00015 

0.999848 

0 

89 

10 

0.020361 

49.114062 

0.020365 

49.103881 

.00021 

0.999793 

50 

20 

0.023269 

42.975713 

0.023275 

42.964077 

.00027 

0.999729 

40 

30 

0.026177 

38.201550 

0.026186 

38.188459 

.00034 

0.999657 

30 

40 

o.  023085. 

34.382316 

0.029097 

34.367771 

.00042 

0.999577 

20 

So 

0.031992 

31.257577 

0.032009 

31.241577 

.00051 

0.999488 

IO 

2 

o 

0.034899 

28.653708 

0.034921 

28.636253 

.00061 

0.999391 

0 

88 

IO 

0.037806 

26.450510 

0.037834 

26.431600 

.00072 

0.999285 

So 

20 

0.040713 

24.562123 

0.040747 

24.541758 

.00083 

0.999171 

40 

30 

0.043619 

22.925586 

0.043661 

22.903766 

.00095 

0.999048 

30 

40 

0.046525 

21.493676 

0.046576 

21.470401 

.00108 

0.998917 

20 

So 

0.049431 

20.230284 

0.049491 

20.205553 

.00122 

0.998778 

10 

3 

0 

0.052336 

19.107323 

0.052408 

19.081137 

.00137 

0.998630 

0 

87 

10 

0.055241 

18.102619 

0.055325 

18.074977 

.00153 

0.998473 

So 

20 

0.058145 

17.198434 

0.058243 

I7.I69337 

.00169 

0.998308 

40 

30 

0.061049 

16.380408 

0.061163 

16.349855 

.00187 

o.998i3S 

30 

40 

0.063952 

15-636793 

0.064083 

15.604784 

.00205 

0.997357 

20 

50 

0.066854 

14.957882 

0.067004 

14.924417 

.00224 

0.997763 

10 

4 

o 

0.069756 

14.335587 

0.069927 

14.300666 

.00244 

0.997564 

O 

86 

IO 

0.072658 

13.763115 

0.072851 

13.726738 

.00265 

0.997357 

50 

20 

0.075559 

13.234717 

0.075776 

13.196888 

.00287 

0.997141 

40 

30 

0.078459 

12.745495 

o  .  078702 

12.706205 

.00309 

0.996917 

30 

40 

0.081359 

12.291252 

0.081629 

12.250505 

.00333 

0.996685 

20 

So 

0.084258 

11.868370 

0.084558 

11.826167 

.00357 

0.996444 

10 

5 

0 

0.087156 

U.4737I3 

0.087489 

11.430052 

.00382 

o.996i95 

0 

85 

10 

0.090053 

11.104549 

0.090421 

11.059431 

.00408 

0.995937 

50 

20 

o  .  092950 

10.758488 

0.093354 

10.711913 

.00435 

0.995671 

40 

30 

o  .  095846 

10.433431 

0.096289 

10.385397 

.00463 

0.995396 

30 

40 

0.098741 

10.127522 

o  .  099226 

10.078031 

.00491 

0-995II3 

20 

So 

0.101635 

9.8391227 

0.102164 

9.7881732 

.00521 

0.994822 

IO 

6 

o 

0.104528 

9.5667722 

o  .  1  051x34 

9.SI43645 

.00551 

0.994522 

0 

84 

10 

o  .  107421 

9.3091699 

0.108046 

9.2553035 

.00582 

0.994214 

50 

20 

0.110313 

9.0651512 

0.110990 

9.0098261 

.00614 

0.993897 

40 

30 

0.113203 

8.8336715 

0.113936 

8.7768874 

.00647 

0.993572 

30 

40 

0.116093 

8.6137901 

0.116883 

8.5555468 

.00681 

0.993238 

20 

So 

0.118982 

8.4045586 

0.119833 

8.3449558 

.00715 

0.992896 

10 

83 

Deg. 

Min. 

Cosine. 

Sec. 

Cotang. 

Tang. 

Cosec. 

Sine. 

Min. 

Deg. 

For  functions  from  83°  10'  to  90°  read  fronvbottom  of  table  upward. 


TRIGONOMETRY 

NATURAL  FUNCTIONS  OF  ANGLES     (Continued) 


195 


Deg. 

Min. 

Sine. 

Cosec. 

Tang. 

Cotang. 

Sec. 

Cosine. 

Min. 

Deg. 

7 

o 

0.121869 

8.2055090 

0.122785 

8.1443464 

.00751 

0.992546 

o 

83 

10 

0.124756 

8.0156450 

0.125738 

7.9530224 

.00787 

0.992187 

So 

,  2Q 

0.127642 

7.8344335 

0.128694 

7.7703506 

.00825 

0.991820 

40 

"  30 

0.130526 

7  .  6612976 

0.131653 

7-5957541 

.00863 

0.991445 

30 

40 

0.133410 

7.4957100 

0.134613 

7.4287064 

.00902 

0.991061 

20 

50 

0.136292 

7-3371909 

O.I37S76 

7.2687255 

.00942 

0.990669 

IO 

8 

0 

O.I39I73 

7-1852965 

0.140541 

7.H53697 

.00983 

0.990268 

0 

88 

10 

0.142053 

7.0396220 

0.143508 

6.9682335 

.01024 

0.989859 

50 

20 

0.144932 

6.8997942 

0.146478 

6.8269437 

.01067 

0.989442 

40 

30 

0.147809 

6.7654691 

O.I4945I 

6.6911562 

.OIIII 

0.989016 

30 

40 

0.150686 

6.6363293 

0.152426 

6.5605538 

.01155 

0.988582 

20 

50 

o.i5356i 

6.5120812 

0.155404 

6.4348428 

.01200 

0.988139 

10 

9 

0 

0.156434 

6.3924532 

0.158384 

6.3I375I5 

.01247 

0.987688 

0 

81 

10 

0.159307 

6.2771933 

0.161368 

6.1970279 

.01294 

0.987229 

50 

20 

0.162178 

6.1660674 

0.164354 

6.0844381 

.01342 

0.986762 

40 

30 

0.165048 

6.0588980 

0.167343 

5.9757644 

.01391 

0.986286 

30 

40 

0.167916 

5.9553625 

0.170334 

5.8708042 

.01440 

0.985801 

20 

50 

0.170783 

S.855392I 

0.173329 

5.7693688 

.01491 

0.985309 

10 

10 

o 

0.173648 

5.7587705 

0.176327 

5.6712818 

.01543 

0.984808 

0 

80 

10 

0.176512 

5.6653331 

0.179328 

5.5763786 

.01595 

0.984298 

50 

20 

0.179375 

5.5749258 

0.182332 

5.4845052 

.01649 

0.983781 

40 

30 

0.182236 

5.4874043 

0.185339 

5.3955172 

.01703 

0.983255 

30 

40 

0.185095 

5.4026333 

0.188359 

5.3092793 

.01758 

0.982721 

20 

So 

0.187953 

5.3204860 

0.191363 

5.2256647 

.01815 

0.982178 

10 

11 

0 

0.190809 

5.2408431 

0.194380 

5.1445540 

.01872 

0.981627 

O 

79 

10 

0.193664 

5.1635924 

0.197401 

5.0658352 

.01930 

0.981068 

So 

20 

0.196517 

5.0886284 

o  .  200425 

4.9894027 

.01989 

o  .  980500 

40 

30 

0.199368 

5.0158317 

0.203452 

4.9I5I570 

.  02049 

0.979925 

30 

40 

0.202218 

4.9451687 

o  .  206483 

4.8430045 

.02110 

0-979341 

20 

So 

0.205065 

4.8764907 

0.209518 

4.7728568 

.02171 

0.978748 

IO 

12 

o 

0.207912 

.8097343 

0.212557 

4.7046301 

.02234 

0.978148 

0 

78 

IO 

0.210756 

.7448206 

0.215599 

4.6382457 

,02298 

0.977539 

50 

20 

0.213599 

.6816748 

0.218645 

4-5736287 

.02362 

0.976921 

40 

30 

0.216440 

.6202263 

0.221695 

4.5107085 

.02428 

0.976296 

30 

40 

0.219279 

.5604080 

0.224748 

4.4494181 

.02494 

0.975662 

20 

50 

0.222116 

.5021565 

0.227806 

4-3896940 

.02562 

0.975020 

10 

13 

0 

0.224951 

4.4454115 

0.230868 

4.3314759 

.  02630 

0-974370 

O 

77 

10 

0.227784 

4.3901158 

0.233934 

4.2747066 

.O27OO 

0.973712 

50 

20 

0.230616 

4.3362150 

0.237004 

4.2193318 

.02770 

0.973045 

40 

30 

0.233445 

4.2836576 

0.240079 

4.1652998 

.02842 

0.972370 

30 

40 

0.236273 

4.2323943 

0.243158 

4.1125614 

.02914 

0.971687 

20 

So 

0.239098 

4.1823785 

0.246241 

4.0610700 

.02987 

0.970995 

10 

76 

Deg. 

Min. 

Cosine. 

Sec. 

Cotang. 

Tang. 

Cosec. 

Sine. 

Min. 

Deg. 

For  functions  from  76°  10'  to  83°  oo'  read  from  bottom  of  table  upward. 


196  PRACTICAL  SURVEYING 

NATURAL  FUNCTIONS  OF  ANGLES     (Continued) 


Deg. 

Min. 

Sine. 

Cosec. 

Tang. 

Cotang. 

Sec. 

Cosine. 

Min. 

Deg. 

14 

0 

0.241922 

4.I33S655 

0.249328 

4  .  0107809 

.03061 

0.970296 

o 

76 

10 

0.244743 

4.0859130 

0.252420 

3.9616518 

.03137 

0.969588 

So 

20 

0.247563 

4-0393804 

0.255517 

3.9136420 

.03213 

0.968872 

40 

30 

0.250380 

3.9939292 

0.258618 

3.8667131 

.03290 

0.968148 

30 

40 

0.253195 

3.9495224 

0.261723 

3.8208281 

.03363 

0.967415 

'20 

50 

0.256008 

3.9061250 

0.264834 

3.7759519 

•03447 

0.966675 

10 

IS 

o 

0.258819 

3.8637033 

0.267949 

3.7320508 

.03528 

0.965926 

O 

75 

10 

0.261628 

3.8222251 

0.271069 

3.6890927 

.03609 

0.965169 

So 

20 

0.264434 

3.7816596 

0.274195 

3.6470467 

.03691 

0.964404 

40 

30 

0.267238 

3.7419775 

0.277325 

3.6058835 

.03774 

0.963630 

30 

40 

0.270040 

3.7031506 

0.280460 

3-5655749 

.03858 

0.962849 

20 

So 

o  .  272840 

3.6651518 

0.283600 

3.5260938 

•03944 

0.962059 

10 

16 

0 

0.275637 

3-6279553 

0.286745 

3.4874144 

.04030 

0.961262 

o 

74 

10 

0.278432 

3.5915363 

0.289896 

3-4495120 

.04117 

0.960456 

So 

20 

0.281225 

3.55587io 

0.293052 

3.4123626 

.04206 

0.959642 

40 

30 

0.284015 

3.5209365 

0.296214 

3-3759434 

.04295 

0.958820 

30 

40 

0.286803 

3.4867110 

0.299380 

3.3402326 

.04385 

0.957990 

20 

So 

0.289589 

3-4531735 

0.302553 

3.3052091 

.04477 

0.957I5I 

IO 

17 

o 

0.292372 

3.4203036 

0.305731 

3.2708526 

.04569 

0.956305 

0 

73 

IO 

0.295152 

3.3880820 

0.308914 

3-2371438 

.04663 

0.955450 

50 

20 

0.297930 

3-3564900 

0.312104 

3.2040638 

.04757 

0.954588 

40 

30 

0.300706 

3-3255095 

0.315299 

3.I7I5948 

.04853 

0.953717 

30 

40 

0.303479 

3.2951234 

0.318500 

3.I397I94 

.04950 

0.952838 

20 

So 

0.306249 

3.2653149 

0.321707 

3.1084210 

.05047 

0.9SI95I 

10 

18 

o 

0.309017 

3.2360680 

0.324920 

3.0776835 

.05146 

0.951057 

0 

72 

10 

0.311782 

3-2073673 

0.328139 

3.04749IS 

.05246 

0.950154 

So 

20 

0.314545 

3.1791978 

0.331364 

3.0178301 

.05347 

0.949243 

40 

30 

0.317305 

3  -I5I5453 

0.334595 

2.9886850 

•05449 

0.948324 

30 

40 

0.320062 

3-1243959 

0.337833 

2  .  9600422 

.05552 

0.947397 

20 

So 

0.322816 

3.0977363 

0.341077 

2.9318885 

.05657 

0.946462 

10 

19 

0 

0.325568 

3.0715535 

0.344328 

2.9042109 

.05762 

0.945519 

o 

71 

10 

0.328317 

3.0458*352 

0.347585 

2.8769970 

.05869 

0.944568 

So 

20 

0.331063 

3.0205693 

0.350848 

2.8502349 

.05976 

0.943609 

40 

30 

0.333807 

2.9957443 

0.354II9 

2.8239129 

.06085 

0.942641 

30 

40 

0.336547 

2.9713490 

0.357396 

2.7980198 

.06195 

0.941666 

20 

50 

0.339285 

2.9473724 

0.360680 

2.7725448 

.06306 

0.940684 

10 

20 

o 

0.342020 

2.9238044 

0.363970 

2.7474774 

.06418 

0.939693 

o 

70 

10 

0.344752 

2  .  9006346 

0.367268 

2  .  7228076 

.06531 

0.938694 

So 

20 

0.347481 

2.8778532 

0.370573 

2.6985254 

.06645 

0.937687 

40 

30 

0.350207 

2.8554510 

0.373885 

2.6746215 

.06761 

0.936672 

30 

40 

0.352931 

2.8334185 

0.377204 

2.6510867 

.06878 

0.935650 

20 

So 

0.355651 

2.8117471 

0.380530 

2.6279I2I 

.06995 

0.934619 

10 

69 

Deg. 

Min. 

Cosine. 

Sec. 

Cotang. 

Tang. 

Cosec. 

Sine. 

Min. 

Deg. 

For  functions  from  69°  10'  to  76°  oo'  read  from  bottom  of  table  upward. 


TRIGONOMETRY 
NATURAL  FUNCTIONS  OF  ANGLES  (Continued) 


197 


Deg. 

Min. 

Sine. 

Cosec. 

Tang. 

Cotang. 

Sec. 

Cosine. 

Min. 

Deg. 

21 

o 

0.358368 

2.7904281 

0.383864 

2.6050891 

.07115 

0.933580 

o 

69 

10 

0.361082 

2.7694532 

0.387205 

2.5826094 

.07235 

0.932534 

50 

20 

0.363793 

2.7488144 

0.390554 

2.5604649 

07356 

0.931480 

40 

30 

0.366501 

2  .  7285038 

0.3939" 

2.5386479 

.07479 

0.930418 

30 

40 

o  .  369206 

2.7085139 

0.397275 

2.5171507 

.07602 

0.929348 

20 

So 

0.371908 

2.6888374 

0.400647 

2.4959661 

.07727 

0.928270 

IO 

22 

o 

0.374607 

2.6694672 

0.404026 

2.4750869 

.07853 

0.927184 

0 

68 

IO 

0-377302 

2  .  6503962 

0.407414 

2.4545061 

.07981 

0.926090 

50 

20 

0.379994 

2.6316180 

0.410810 

2.4342172 

.08109 

0.924989 

40 

30 

0.382683 

2.6I3I259 

0.414214 

2.4142136 

.08239 

0.923880 

30 

40 

0.385369 

2.5949137 

0.417626 

2.3944889 

.08370 

0.922762 

20 

50 

0.388052 

2.5769753 

0.421046 

2.3750372 

.08503 

0.921638 

IO 

23 

o 

0.390731 

2.5593047 

0.424475 

2.3558524 

.08636 

0.920505 

0 

67 

10 

0.393407 

2.5418961 

0.427912 

2.3369287 

.08771 

0.919364 

50 

20 

0.396080 

2.5247440 

0.431358 

2.3182606 

.08907 

0.918216 

40 

30 

0.398749 

2.5078428 

0.434812 

2.2998425 

.09044 

0.917060 

30 

40 

0.401415 

2.49II874 

0.438276 

2.2816693 

.09183 

0.915896 

20 

So 

o  .  404078 

2.4747726 

0.441748 

2.2637357 

.09323 

0.914725 

10 

24 

0 

0.406737 

2.4585933 

0.445229 

2.2460368 

.09464 

0.913545 

o 

66 

10 

0.409392 

2.4426448 

0.448719 

2.2285676 

.09606 

0.912358 

50 

20 

0.412045 

2.4269222 

0.452218 

2.2113234 

.09750 

0.911164 

40 

30 

0.414693 

2.4II42IO 

0.455726 

2.1942997 

.09895 

0.909961 

30 

ii 

40 

0.4I733S 

2.3961367 

0.459244 

2.1774920 

.10041 

0.908751 

20 

50 

0.419980 

2.3810650 

0.462771 

2.1608958 

.  10189 

0.907533 

IO 

25 

o 

0.422618 

2.3662016 

0.466308 

2.1445069 

-10338 

0.906308 

o 

66 

10 

0.425253 

2.3515424 

0.469854 

2.1283213 

.10488 

0.905075 

50 

20 

0.427884 

2.3370833 

0.473410 

2.1123348 

.10640 

0.903834 

40 

30 

0.430511 

2  .  3228205 

0.476976 

2.0965436 

.  10793 

0.902585 

30 

40 

0.433135 

2.3087501 

0.480551 

2.0809438 

.10947 

0.901329 

20 

50 

0.435755 

2.2948685 

0.484137 

2.0655318 

.11103 

0.900065 

10 

26 

o 

0.438371 

2.28II72O 

0.487733 

2  .  0503038 

.11260 

0.898794 

0 

64 

10 

0.440984 

2.2676571 

0.491339 

ft.0352565 

.11419 

0.897515 

50 

20 

0.443593 

2  .  2543204 

0.494955 

2  .  O203862 

.11579 

0.896229 

40 

30 

0.446198 

2.2411585 

0.498582 

2.0056897 

.11740 

0.894934 

30 

40 

0.448799 

2.228l68l 

0.502219 

.9911637 

•11903 

0.893633 

20 

50 

0.451397 

2.2153460 

0.505867 

.9768050 

.12067 

0.892323 

10 

27 

o 

0.453990 

2  .  2O26893 

0.509525 

.9626105 

.12233 

0.891007 

o 

63 

IO 

0.456580 

2.I90I947 

0.5I3I95 

.9485772 

.12400 

0.889682 

50 

20 

0.459166 

2.1778595 

0.516876 

.9347020 

.12568 

0.888350 

40 

30 

0.461749 

2.1656806 

0.520567 

.9209821 

.12738 

0.887011 

30 

40 

0.464327 

2.I536S53 

0.524270 

.9074147 

.12910 

0.885664 

20 

50 

0.466901 

2.I4I7808 

0.527984 

.8939971 

.13083 

0.884309 

10 

62 

Deg. 

Min. 

Cosine. 

Sec. 

Cotang. 

Tang. 

Cosec. 

Sine. 

Min. 

Deg. 

For  functions  from  62°  10'  to  69°  oo'  read  from  bottom  of  table  upward. 


198  PRACTICAL  SURVEYING 

NATURAL  FUNCTIONS  OF  ANGLES  (Continued} 


Deg. 

Min. 

Sine. 

Cosec. 

Tang. 

Cotang. 

Sec. 

Cosine. 

Min. 

Deg. 

28 

o 

0.469472 

2.1300545 

0.531709 

.8807265 

.13257 

0.882948 

o 

62 

10 

0.472038 

-1184737 

0.535547 

.  8676003 

.  13433 

0.881578 

So 

20 

0.474600 

.  1070359 

0.539I9S 

.8546159 

.13610 

0.880201 

40 

30 

0.477159 

.0957385 

0.542956 

.8417409 

•  13789 

0.878817 

30 

40 

0.479713 

.0845792 

0.546728 

.8290628 

•  13970 

0.877425 

20 

50 

0.482263 

.0735556 

0.550515 

.8164892 

.14152 

0.876026 

10 

29 

0 

0.484810 

.0626653 

0.554309 

.8040478 

•  14335 

0.874620 

0 

61 

IO 

0.487352 

.0519061 

0.558118 

.7917362 

.14521 

0.873206 

So, 

20 

0.489890 

.0412757 

0.561939 

•7795524 

•  14707 

0.871784 

40 

30 

0.492424 

•  0307720 

0.565773 

.7674940 

.14896 

0.870356 

30 

40 

0.494953 

.  0203929 

0.569619 

.  7555590 

•  15085 

0.868920 

20 

50 

0.497479 

.0101362 

0.573478 

•7437453 

•  15277 

0.867476 

10 

30 

o 

0.500000 

.0000000 

0.577350 

.7320508 

•  15470 

o  .  866025 

0 

60 

IO 

0.502517 

.9899822 

0.581235 

.7204736 

.15665 

0.864567 

50 

20 

0.505030 

.9800810 

0.585134 

.7090116 

.  15861 

0.863102 

40 

30 

0.507538 

.9702944 

0.589045 

.6976631 

.  16059 

0.861629 

30 

40 

0.510043 

.9606206 

0.592970 

.6864261 

.  16259 

0.860149 

20 

50 

0.512543 

.9510577 

0.596908 

.6752988 

.  16460 

0.858662 

IO 

31 

0 

0.515038 

.9416040 

0.600861 

.6642795 

.16663 

0.857167 

0 

59 

IO 

0.517529 

.9322578 

0.604827 

.6533663 

.16868 

0.855665 

50 

20 

0.520016 

.9230173 

0.608807 

.6425576 

.17075 

0.854156 

40 

30 

0.522499 

.9138809 

0.612801 

.6318517 

.17283 

0.852640 

30 

40 

0.524977 

.9048469 

0.616809 

.6212469 

•17493 

0.851117 

20 

50 

0.527450 

.8959138 

0.620832 

.6107417 

.  17704 

0.849586 

IO 

32 

o 

0.529919 

.8870799 

0.624869 

.6003345 

.17918 

0.848048 

O 

68 

10 

0.532384 

.8783438 

0.628921 

.5900238 

.18133 

o  .  846503 

50 

20 

0.534844 

.8697040 

0.632988 

.5798079 

.  18350 

0.844951 

40 

30 

0.537300 

.8611590 

0.637079 

.5696856 

.18569 

0.843391 

30 

40 

0.539751 

.8527073 

0.641167 

.5596552 

.18790 

0.841825 

20 

50 

0.542197 

.8443476 

0.645280 

•5497155 

.19012 

0.840251 

10 

33 

O 

0.544639 

.8360785 

0.649408 

.5398650 

.  19236 

0.838671 

o 

67 

10 

0.547076 

.8278985 

0.653531 

.5301025 

.  19463 

0.837083 

So 

20 

0.549509 

.8198065 

«o.  657710 

.5204261 

.  19691 

0.835488 

40 

30 

0.551937 

.8118010 

0.661886 

.5108352 

.19920 

0.833886 

30 

40 

o.55436o 

.8038809 

0.666077 

.5013282 

.20152 

0.832277 

20 

So 

0.556779 

.7960449 

0.670285 

.4919039 

.20386 

0.830661 

10 

34 

0 

0.559193 

.  7882916 

0.674509 

.4825610 

.20622 

0.829038 

O 

66 

IO 

0.561602 

.7806201 

0.678749 

.4732983 

.20859 

0.827407 

So 

20 

0.564007 

.7730290 

0.683007 

.4641147 

.21099 

o  .  825770 

40 

30 

o  .  566406 

.  7655173 

0.687281 

.4550090 

.21341 

0.824126 

30 

40 

0.568801 

.7580837 

0.691573 

.4459801 

.21584 

0.822475 

20 

50 

0.57H9I 

.7507273 

0.695881 

.4370268 

.21830 

0.820817 

10 

55 

Deg. 

Min. 

Cosine. 

Sec. 

Cotang. 

Tang. 

COGCC. 

Sine. 

Min. 

Deg. 

For  functions  from  55°  10'  to  62°  oo'  read  from  bottom  of  table  upward. 


TRIGONOMETRY 
NATURAL  FUNCTIONS  OF  ANGLES  (Continued) 


199 


Deg. 

Min. 

Sine. 

Cosec. 

Tang. 

Cotang. 

Sec. 

Cosine. 

Min. 

Deg. 

35 

o 

0.573576 

.7434468 

o  .  700208 

.4281480 

1.22077 

0.819152 

o 

66 

10 

o:57S957 

.7362413 

0.704552 

.4193427 

1.22327 

0.817480 

50 

20 

0.578332 

.  7291096 

0.708913 

.4106098 

1.22579 

0.815801 

40 

30 

0.580703 

.  7220508 

0.713293 

.4019483 

1.22833 

0.814116 

30 

40 

0.583069 

.7150639 

0.717691 

-3933571 

1.23089 

0.812423 

20 

So 

0.585429 

.  7081478 

0.722108 

.3848355 

1.23347 

0.810723 

10 

36 

o 

0.587785 

.  7013016 

0.726543 

.3763810 

1.23607 

0.809017 

o 

64 

10 

0.590136 

.6945244 

0.730996 

-3679959. 

1.23869 

0.807304 

50 

20 

0.592482 

.6878151 

0.735469 

.3596764 

1.24134 

0.805584 

40 

30 

0.594823 

.6811730 

o.73996i 

.3514224 

1.24400 

0.803857 

30 

40 

0.597IS9 

.6745970 

0.744472 

.3432331 

1.24669 

0.802123 

20 

50 

0.599489 

.6680864 

0.749003 

-3351075 

1.24940 

0.800383 

IO 

37 

o 

0.601815 

.6616401 

0.753554 

.3270448 

1.25214 

0.798636 

0 

53 

IO 

0.604136 

.6552575 

0.758125 

.3190441 

1.25489 

0.796882 

50 

20 

0.606451 

.6489376 

0.762716 

.3111046 

1-25767 

0.795I2I 

40 

30 

0.608761 

.6426796 

0.767627 

.3032254 

i  .  26047 

0-793353 

30 

40 

0.611067 

.6364828 

0.771959 

.2954057 

1.26330 

0.791579 

20 

50 

0.613367 

.6303462 

0.776612 

.2876447 

1.26615 

0.789798 

10 

38 

0 

0.615661 

.6242692 

0.781286 

.2799416 

1.26902 

0.788011 

o 

62 

10 

0.617951 

.6182510 

0.785981 

.2722957 

1.27191 

0.786217 

50 

20 

0.620235 

.6122908 

0.790698 

.2647062 

1.27483 

0.784416 

40 

30 

0.622515 

.6063879 

0.795436 

•2571723 

1.27778 

0.782608 

30 

40 

0.624789 

.6005416 

0.800196 

.2496933 

1.28075 

0.780794 

20 

So 

o  .  627057 

•  59475" 

0.804080 

.2422685 

1.28374 

0.778973 

IO 

39 

o 

0.629320 

.5890157 

0.809784 

.2348972 

1.28676 

0.777146 

o 

61 

IO 

0.631578 

.5833318 

0.814612 

.2275786 

1.28980 

0.775312 

50 

20 

0.633831 

-5777077 

0.819463 

.2203121 

1.29287 

0.773472 

40 

30 

0.63*078 

.5721337 

0.824336 

.2130970 

1.29597 

0.771625 

30 

40 

0.638320 

.5666121 

0.829234 

.2059327 

1.29909 

0.769771 

20 

So 

0.640557 

.5611424 

0.834155 

.1988184 

1.30223 

0.767911 

IO 

40 

o 

0.642788 

.5557238 

0.839100 

.1917536 

1.30541 

0.766044 

o 

60 

IO 

o  .  645013 

.5503558 

0.844069 

•  1847376 

1.30861 

0.764171 

50 

20 

0.647233 

.5450378 

0.849062 

.1777698 

1.31183 

0.762292 

40 

30 

0.649448 

.5397690 

0.854081 

.  1708496 

1-31509 

o  .  760406 

30 

40 

0.651657 

-5345491 

0.859124 

.  1639763 

1.31837 

0.758514 

20 

50 

0.653861 

.5293773 

0.864193 

.1571495 

1.32168 

0.756615 

10 

41 

o 

0.656059 

.5242531 

0.869287 

.1503684 

1.32501 

0.754710 

0 

49 

IO 

0.658252 

.5191759 

0.874407 

.  1436326 

1.32838 

0.752798 

50 

20 

0.660439 

.5141452 

0.879553 

.  1369414 

I.33I77 

0.750880 

40 

30 

0.662620 

.5091605 

0.884725 

.  1302944 

I.335I9 

0.748956 

30 

40 

0.664796 

.5042211 

0.889924 

.1236909 

1.33864 

0.747025 

20 

So 

0.666966 

.4993267 

0.895151 

.1171305 

1.34212 

0.745088 

IO 

48 

Deg. 

Min. 

Cosine. 

Sec. 

Cotang. 

Tang. 

Cosec. 

Sine. 

Min. 

Deg. 

For  functions  from  48°  10'  to  55°  oo'  read  from  bottom  of  table  upward. 


200  PRACTICAL   SURVEYING 

NATURAL  FUNCTIONS  OF  ANGLES  (Continued) 


Deg. 

Min. 

Sine. 

Cosec. 

Tang. 

Cotang. 

Sec. 

Cosine. 

Min. 

Deg. 

42 

0 

0.669131 

.4944765 

0.900404 

.1106125 

.34563 

0.743145 

o 

48 

10 

0.671289 

.4896703 

0.905685 

.  1041365 

.34917 

0.741195 

So 

20 

0.673443 

.4849073 

0.910994 

.  0977020 

•35274 

0.739239 

40 

30 

0.675590 

.4801872 

0.916331 

.0913085 

.35634 

0.737277 

30 

40 

0.677732 

.4755095 

0.921697 

.0849554 

•35997 

c.735309 

20 

50 

0.679868 

•4708736 

0.927091 

.0786423 

.36363 

0.733335 

10 

43 

o 

0.681998 

.4662792 

0.932515 

.0723687 

.36733 

0.731354 

o 

47 

IO 

0.684123 

.4617257 

0.937968 

.0661341 

•37105 

0.729367 

So 

£0 

0.686242 

.4572127 

0.943451 

.0599381 

.37481 

0.727374 

40 

30 

0.688355 

•4527397 

0.948965 

.0537801 

.37860 

0.725374 

30 

40 

0.690462 

.4483063 

o.9545o8 

.0476598 

.38242 

0.723369 

20 

So 

0.692563 

.4439120 

0.960083 

.0415767 

.38628 

0.721357 

IO 

44 

o 

0.694658 

.4395565 

0.965689 

.0355303 

.39016 

0.719340 

o 

46 

IO 

0.696748 

.4352393 

0.971326 

.0295203 

.39409 

0.717316 

50 

20 

0.698832 

.4309602 

0.976996 

.0235461 

.39804 

0.715286 

40 

30 

0.700909 

.4267182 

0.982697 

.0176074 

.40203 

0.713251 

30 

40 

0.702981 

.4225134 

0.988432 

.0117088 

.40606 

0.711209 

20 

50 

0.705047 

.4183454 

0.994199 

.0058348 

.41012 

0.709161 

10 

45 

0 

0.707107 

1.4142136 

I.OOOOOO 

I.  0000000 

1.41421 

0.707107 

0 

45 

Deg. 

Min. 

Cosine. 

Sec. 

Cotang. 

Tang. 

Cosec. 

Sine. 

Min. 

Deg. 

For  functions  from  45°  o'  to  48°  oo'  read  from  bottom  of  table  upward. 


TRIGONOMETRY 
LOGARITHMS  OF  NUMBERS  FROM  o  TO  1000 


201 


No. 

o 

I 

2 

3 

4 

5 

6 

7 

8 

9 

0 

o 

ooooo 

30103 

47712 

/•___/- 

OO2OO 

69897 

77815 

84510 

90309 

95424 

10 

ooooo 

00432 

00860 

01284 

01703 

02119 

02531 

02938 

03342 

03743 

II 

04139 

04532 

04922 

05308 

05690 

06070 

06446 

06819 

07188 

07555 

12 

07918 

08279 

08636 

08991 

09342 

09691 

10037 

10380 

10721 

11059 

13 

1  1394 

11727 

12057 

12385 

12710 

13033 

13354 

13672 

13988 

14301 

14 

14613 

14922 

15229 

15534 

15836 

16137 

16435 

16732 

17026 

I73I9 

IS 

17609 

17898 

18184 

18469 

18752 

19033 

19312 

19590 

19866 

20140 

16 

20412 

20683 

20952 

21219 

21484 

21748 

220II 

22272 

22531 

22789 

17 

23045 

23300 

23553 

23805 

24055 

24304 

24551 

24797 

25042 

25285 

18 

25527 

25768 

26007 

26245 

26482 

26717 

26951 

27184 

27416 

27646 

19 

27875 

28103 

28330 

28556 

28780 

29003 

29226 

29447 

29667 

29885 

20 

30103 

30320 

30535 

30750 

30963 

3H75 

31387 

31597 

31806 

32015 

21 

32222 

32428 

32634 

32838 

33041 

33244 

33445 

33646 

33846 

34044 

22 

34242 

34439 

34635 

34830 

35025 

35218 

354H 

35603 

35793 

35984 

23 

36173 

36361 

36549 

36736 

36922 

37107 

37291 

37475 

37658 

37840 

24 

38021 

38202 

38382 

38561 

38739 

38917 

39094 

39270 

39445 

39620 

25 

39794 

39967 

40140 

40312 

40483 

40654 

40824 

40993 

41162 

41330 

26 

41497 

41664 

41830 

41996 

42160 

42325 

42488 

42651 

42813 

42975 

27 

43136 

43297 

43457 

43616 

43775 

43933 

44091 

44248 

44404 

4456o 

28 

44716 

44871 

45025 

45179 

45332 

45484 

45637 

45788 

45939 

46090 

29 

46240 

46389 

46538 

46687 

46835 

46982 

47129 

47276 

47422 

47567 

30 

47712 

47857 

48001 

48144 

48287 

48430 

48572 

48714 

48855 

48996 

31 

49136 

49276 

49415 

49554 

49693 

49831 

49969 

50106 

50243 

50379 

32 

50515 

50651 

50786 

50920 

51055 

51188 

51322 

51455 

51587 

51720 

33 

5I85I 

SI983 

52H4 

52244 

52375 

52504 

52633 

52763 

52892 

53020 

34 

53148 

53275 

53403 

53529 

53656 

53782 

53908 

54033 

54158 

54283 

35 

54407 

54531 

54654 

54777 

54900 

55023 

55145 

55267 

55388 

55509 

36 

55630 

55751 

55871 

55991 

56110 

56229 

56348 

56467 

56585 

56703 

37 

56820 

56937 

57054 

57I7I 

57287 

57403 

57519 

57634 

57749 

57864 

38 

57978 

58093 

58206 

58320 

58433 

58546 

58659 

58771 

58883 

58995 

39 

59io6 

592i8 

59329 

59439 

59550 

59660 

59770 

59879 

59988 

60097 

40 

60206 

60314 

60423 

60531 

60638 

60746 

6o853 

60959 

61066 

61172 

41 

61278 

61384 

61490 

61595 

61700 

61805 

61909 

62014 

62118 

62221 

42 

62325 

62428 

62531 

62634 

62737 

62839 

62941 

63043 

63144 

63246 

43 

63347 

63448 

63548 

63649 

63749 

63849 

63949 

64048 

64147 

64246 

44 

64345 

64444 

64542 

64640 

64738 

64836 

64933 

65031 

65128 

65225 

45 

65321 

65418' 

65514 

65610 

65706 

65801 

65896 

65992 

66087 

66181 

46 

66276 

66370 

66464 

66558 

66652 

66745 

66839 

66932 

67025 

67117 

47 

67210 

67302 

67394   67486 

67578 

67669 

67761 

67852 

67943 

68034 

48 

68124 

68215 

68305   68395 

68485 

68574 

68664 

68753 

68842 

68931 

49 

69020 

69108 

69197 

69285 

69373 

69461 

69548 

69636 

69723 

69810 

50 

69897 

69984 

70070 

70157 

70243 

70329 

70415 

70501 

70586 

70672 

Si 

70757 

70842 

70927 

71012 

71096 

71181 

71265 

71349 

71433 

7I5I7 

52 

71600 

71684 

71767 

71850 

71933 

72016 

72099 

72181 

72263 

72346 

53 

72428 

72509 

72591 

72673 

72754 

72835 

72916 

72997 

73078 

73159 

54 

73239 

73320 

73400 

7348o 

7356o 

73640 

73719 

73799 

73878 

73957 

202  PRACTICAL   SURVEYING 

LOGARITHMS  OF  NUMBERS  FROM  o  TO  1000  (Continued] 


No. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

55 

74036 

74H5 

74194 

74273 

74351 

74429 

74507 

74586 

74663 

74741 

56 

748i9 

74896 

74974 

75051 

75128 

75205 

75282 

75358 

75435 

7551  1 

57 

75587 

75664 

75740 

75815 

75891 

75967 

76042 

76118 

76193  '  76268 

58 

76343 

76418 

76492 

76567 

76641 

76716 

76790 

76864 

76938  i  77012 

59 

77085 

77159 

77232 

77305 

77379 

77452 

77525 

77597 

77670 

77743 

60 

77815 

77887 

77960 

78032 

78104 

78176 

78247 

78319 

78390 

78462 

61 

78533 

78604 

78675 

78746 

78817 

78888 

78958 

79029 

79099   79169 

62 

79239 

79309 

79379 

79449 

79518 

79588 

79657 

79727 

79796   79865 

63 

79934 

80003 

80072 

80140 

80209 

80277 

80346 

80414 

80482  ,  80550 

64 

80618 

80686 

80754 

80821 

80889 

80956 

81023 

81090 

81158   81224 

65 

81291 

81358 

81425 

81491 

81558 

81624 

81690 

81756 

81823   81889 

66    81954 

82020 

82086 

82151 

82217 

82282 

82347 

82413 

82478   82543 

67 

82607 

82672 

82737 

82802 

82866 

82930 

82995 

83059 

83123   83187 

68 

83251 

83315 

83378 

83442 

83506 

83569 

83632 

83696 

83759   83822 

69 

83885 

83948 

84011 

84073 

84136, 

84198 

84261 

84323 

84386 

84448 

70 

84510 

84572 

84634 

84696 

84757 

84819 

84880 

84942 

85003 

85065 

7i 

85126 

85187 

85248 

85309 

85370 

85431 

85491 

85552 

85612   85673 

72 

85733 

85794 

85854 

853I4 

85974 

86034 

86094 

86153 

86213   86273 

73 

86332 

86392 

86451 

86510 

86570 

86629 

86638 

86747 

86806 

86864 

74 

86923 

86982 

87040 

87093 

87157 

87216 

87274 

87332 

87390 

87448 

75 

87506 

87564 

87622 

87680 

87737 

87795 

87852 

87910 

87967 

88024 

76 

88081 

88138 

88196 

88252  |  88309 

88366 

88423 

88480 

88536 

88593 

77 

88649 

88705 

88762 

88818  i  88874 

88930 

88986 

89042 

89098 

89154 

78 

89209 

89265 

89321 

89376 

89432 

89487 

89542 

89597 

89653  !  89708 

79 

89763 

89818 

89873 

89927 

89982 

90037 

90091 

90146 

90200 

90255 

80 

90309 

90363 

90417 

90472 

90526 

90580 

90634 

90687 

90741 

90795 

81 

90849 

90902 

90956 

91009 

91062   91116 

91169 

91222 

91275 

91328 

82 

9I38I 

91434 

91487 

91540 

91593   91645 

91698 

9I75I 

91803 

91855 

83 

91908 

91960 

92012 

92065 

92117   92169 

92221 

92273 

92324 

92376 

84 

92428 

92480 

92531 

92583 

92634   92686 

92737 

92788 

92840 

92891 

85 

92942 

92993 

93044 

93095 

93146   93197 

93247 

93298 

93349 

93399 

86 

93450 

935QO 

93551 

936oi 

93651   93702 

93752 

93802 

93852 

93902 

87 

93952 

94002 

94052 

94IOI 

94151   94201 

94250 

94300 

94349 

94399 

88 

94448 

94498 

94547 

94596 

94645   946Q4 

94743 

94792 

94841 

94890 

89 

94939 

94988 

95036 

95085 

95134 

95182 

95231 

95279 

95328 

95376 

90 

95424 

95472 

95521 

95569 

95617 

95665 

95713 

9576i 

95809 

95856 

9i 

95904 

95952 

95999 

96047 

96095   96142 

96190 

96237 

96284 

96332 

92 

96379 

96426 

96473 

96520 

96567   96614 

96661 

96708 

96755 

96802 

93 

96848 

96895 

96942 

96988 

97035   97081 

97128 

97174 

97220 

97267 

94 

97313 

97359 

97405 

97451 

97497   97543 

97589 

97635 

97681 

97727 

95 

97772 

978i8 

97864 

97909 

97955  ,  98000 

98046 

98091 

98137 

98182 

96 

98227 

98272 

98318 

98363 

98408 

98453 

98498 

98543 

98588 

98632 

97 

98677 

98722 

98767 

98811 

98856 

98900 

98945 

98989 

99034 

99078 

98 

99123 

99167 

9921  1 

99255 

99300 

99344 

99388 

994.V 

99476 

99520 

99 

99564 

99607 

99651 

99695 

99739 

99782 

99826 

99870 

99913 

99957 

TRIGONOMETRY 
LOGARITHMIC  FUNCTIONS  OF  ANGLES 


203 


Deg. 

Min. 

Sine. 

Cosec. 

Tang. 

Cotang. 

Sec. 

Cosine. 

Min. 

Deg. 

0 

0 

—  oc 

OC 

OC 

OC 

lO.OOOOO 

IO.  OOOOO 

o 

90 

10 

7.46373 

12.53627 

7.46373 

12.53627 

IO.OOOOO 

9.99999 

So 

20 

7  •  76475 

12.23525 

7.76476 

12.23524 

10.00000 

9-99999 

40 

30 

7.94084 

12.05916 

7.94086 

12.05914 

IO.OOO02 

9.99998 

30 

40 

8.06578 

11.93422 

8.06581 

11.93419 

10.00003 

9-99997 

20 

50 

8.16268 

11.83732 

8.16272 

II..83727 

10.00005 

9-99995 

10 

1 

0 

8.24186 

11.75814 

8.24192 

11.75808 

10.00007 

9-99993 

0 

89 

10 

8.30879 

11.69121 

8.30888 

11.69111 

10.00009 

9-99991 

50 

20 

8.36678 

11.63322 

8.36689 

11.63311 

IO.OOOI2 

9-99988 

40 

30 

8.41792 

11.58208 

8.41807 

II.58I93 

10.00015 

9-99985 

30 

40 

8.46366 

11.53634 

8.46385 

II.536I5 

10.00018 

9.99982 

20 

So 

8.50504 

11.49496 

8.50527 

11-49473 

IO.OO022 

9-99978 

IO 

2 

0 

8.54282 

II.457I8 

8.54308 

11.45692 

IO.OOO26 

9-99974 

0 

88 

IO 

8.57757 

11.42243 

8.57788 

11.42212 

10.00031 

9.99969 

50 

20 

8.60973 

11.39027 

8.61009 

11.38990 

10.00036 

9.99964 

40 

30 

8.63968 

11.36032 

8.64009 

II-3599I 

10.00041 

9  99959 

30 

40 

8.66769 

II.3323I 

8.66816 

II.33I84 

IO.OOO47 

9-99953 

20 

50 

8.69400 

11.30600 

8.69453 

H.30547 

10.00053 

9-99947 

IO 

a 

0 

8.71880 

11.28120 

8.71940 

11.28060 

IO.OOO6O 

9-99940 

o 

87 

IO 

8.74226 

n.25774 

8.74292 

11.25708 

10.00066 

9-99934 

So 

20 

8.76451 

H.23549 

8.76525 

11.23475 

IO.OOO74 

9-99926 

40 

30 

8.78567 

11.21432 

8.78648 

11.21351 

10.00081 

9.99919 

30 

40 

8.80585 

11.19415 

8.80674 

11.19326 

10.00089 

9-999H 

20 

50 

8.82513 

11.17487 

8.82610 

11.17390 

10.00097 

9.99903 

IO 

4 

0 

8.84358 

11.15642 

8.84464 

11.15536 

IO.OOI06 

9.99894 

o 

86 

10  ' 

8.86128 

11.13872 

8.86243 

II  13757 

IO.OOII5 

9-99885 

50 

20 

8.87829 

11.12171 

8.87953 

11.12047 

IO.OOI24 

9-99876 

40 

30 

8.89464 

11.10536 

8.89598 

11.10402 

10.00134 

9.99866 

30 

40 

8.91040 

11.08960 

8.91185 

11.08815 

IO.OOI44 

9.99856 

20 

50 

8.92561 

11.07439 

8.92716 

11.07284 

10.00155 

9.99845 

10 

5 

0 

8.04030 

11.05970 

•8.94195 

11.05805 

IO.OOI66 

9.99834 

o 

85 

10 

8.95450 

11.04550 

8.95627 

n.04373 

IO.OOI77 

9  99823 

50 

20 

8.96825 

11-03175 

8-97013 

11.02987 

10.00188 

9.99812 

40 

30 

8.03157 

11.01843 

8.98358 

11.01642 

10  .  OO2OO 

9.998oo 

30 

40 

8.99450 

I  I  .  00550 

8.99662 

11.00337 

IO.OO2I3 

9.99787 

20 

50 

9.00704 

10.99296 

9.00930 

10.99070 

IO.OO225 

9-99775 

10 

6 

o 

9  01923 

10.98076 

9.02162 

10.97838 

10.00239 

9-9976i 

0 

84 

IO 

9.03109 

10.96891 

9  03361 

10.96639 

10.00252 

9-99748 

50 

20 

9  .  04262 

10.95738 

•9-04528 

10.95472 

10.00266 

9-99734 

40 

30 

9  05386 

10.94614 

9.05666 

10.94334 

10.00280 

9-99720 

30 

40 

9.06480 

10.93519 

9-06775 

10.93225 

IO.OO295 

9-99705 

20 

50 

9.07548 

10.92452 

9.07858 

10.92142 

IO.O03IO 

9.99690 

10 

83 

Cosine. 

Sec. 

Cotang. 

Tang.    Cosec. 

Sine. 

I 

For  functions  from  83°  10'  to  90°  oc/  read  from  bottom  of  table  upward. 


204  PRACTICAL   SURVEYING 

LOGARITHMIC  FUNCTIONS  OF  ANGLES  (Continued) 


Deg. 

Min. 

Sine. 

Cosec. 

Tang. 

Cotang. 

Sec. 

Cosine. 

Min. 

Deg. 

7 

o 

9-08589 

10.91411 

9.08914 

10.91086 

10.00325 

9.99675 

0 

83 

IO 

9.09606 

10.90394 

9.09947 

10.90053 

10.00341 

9.99659 

So 

20 

9-10599 

10.89401 

9-10956 

10.89044 

10.00357 

9.99643 

40 

30 

9.11570 

10.88430 

9-II943 

10.88057 

10.00373 

9.99627 

30 

40 

9.12519 

10.87481 

9-12909 

10.87091 

10.00390 

9.99610 

20 

So 

9-13447 

10.86553 

9.13854 

10.86146 

10.00407 

9  99593 

10 

8 

o 

9.14356 

10.85644 

9.14780 

10.85220 

10.00425 

9  99575 

0 

82 

IO 

9.15245 

10.84755 

9.15688 

10.84312 

10.00443 

9-99557 

50 

20 

9.16116 

10.83884 

9.16577 

10.83423 

10.00461 

9-99539 

40 

30 

9.16970 

10.83030 

9-17450 

10.82550 

10.00480 

9-99520 

30 

40 

9.17807 

10.82193 

9.18306 

10.81694 

10.00499 

9-99501 

20 

50 

9.18628 

10.81372 

9.19146 

10.80854 

10.00518 

9.99482 

IO 

9 

O 

9-19433 

10.80567 

9.19971 

10.80029 

10.00538 

9.99462 

o 

81 

IO 

9.20223 

10.79777 

9.20782 

10.79218 

10.00558 

9.99442 

50 

20 

9.20999 

10.79001 

9-21578 

10.78422 

10.00579 

9.99421 

40 

30 

9.21761 

10.78239 

9.22361 

10.77639 

10.00600 

9.99400 

30 

40 

9.22509 

10.77491 

9-23130 

10.76870 

10.00621 

9-99379 

20 

50 

9-23244 

10.76756 

9-23887 

10.76113 

10.00643 

9-99357 

10 

10 

0 

9-23967 

10.76033 

9-24632 

10.75368 

10.00665 

9-99335 

0 

80 

IO 

9-24677 

10.75323 

9-25365 

10.74635 

10.00687 

9-99313 

50 

20 

9.25376 

10.74624 

9.26086 

10.73914 

10.00710 

9.99290 

40 

30 

9.26063 

10.73937 

9.26797 

10.73203 

10.00733 

9.99267 

30 

40 

9.26739 

10.73261 

9.27496 

10.72503 

10.00757 

9-99243 

20 

50 

9-27405 

10.72595 

9.28186 

10.71814 

10.00781 

9.99219 

IO 

11 

0 

9.28060 

10.71940 

9.28865 

10,71135 

10.00805 

9.99195 

o 

79 

10 

9.28705 

10.71295 

9-29535 

10.70465 

10.00830 

9.99170. 

50 

20 

9-29340 

10.70660 

9.30195 

10.69805 

10.00855 

9-99M5 

40 

30 

9-29966 

10.70034 

9-30846 

10.69154 

10.00881 

9.99119 

30 

40 

9-30582 

10.69418 

9.31489 

10.68511 

10.00901 

9-99093 

20 

50 

9.3"89 

10.68811 

9.32122 

10.67878 

10.00933 

9.99067 

IO 

12 

o 

9.31788 

10.68212 

9.32747 

10.67253 

10.00960 

9.99040 

O 

78 

10 

9-32378 

10.67622 

9.33365 

10.66635 

10.00987 

9.99013 

50 

20 

9.32960 

10.67040 

9-33974 

10.66026 

10.01014 

9.98986 

40 

30 

9-33534 

10.66466 

9.34576 

10.65424 

10.01042 

9.98958 

30 

40 

9-34100 

10.65900 

9.35170 

10.64830 

10.01070 

9.98930 

20 

50 

9.34658 

10.65342 

9-35757 

10.64243 

10.01099 

9.98901 

IO 

13 

o 

9.35209 

10.64791 

9.36336 

10.63664 

10.01128 

9.98872 

o 

77 

IO 

9-35752 

10.64248 

9-36909 

10.63091 

10.01157 

9.98843 

So 

20 

9.36289 

10.63711 

9.37476 

10.62524 

10.01187 

9-98813 

40 

30 

9-36819 

10.63181 

9-38035 

10.61965 

10.01217 

9.98783 

30 

40 

9-37341 

10.62659 

9-38589 

10.61411 

10.01247 

9.98752 

20 

50 

9.37858 

10.62142 

9-39r36 

10.60864 

10.01278 

9.98722 

IO 

76 

Cosine. 

Sec. 

Cotang. 

Tang. 

Cosec. 

Sine. 

For  functions  from  76°  10'  to  83°  oo'  read  from  bottom  of  table  upward. 


TRIGONOMETRY 
LOGARITHMIC  FUNCTIONS  OF  ANGLES  (Continued) 


205 


Deg. 

Min. 

Sine. 

Cosec. 

Tang. 

Cotang. 

Sec. 

Cosine. 

Min. 

Deg. 

14 

0 

9-38368 

10.61632 

9-39677 

10.60323 

10.01310 

9.98690 

0 

76 

IO 

9-38871 

10.61129 

9.40212 

10.59788 

10.01341 

9.98659 

So 

20 

9.39369 

10.60631 

9.40742 

10.59258 

10.01373 

9.98627 

40 

30 

9.3986o 

10.60140 

9.41266 

10.58734 

10.01406 

9.98594 

30 

40 

9.40346 

10.59654 

9.41784 

10.58216 

10.01439 

9.98561 

20 

50 

9.40825 

io.59i75 

9.42297 

10.57703 

10.01472 

9-98528 

IO 

15 

0 

9.41300 

10.587* 

9.42805 

10.57195 

10.01506 

9.98494 

o 

75 

10 

9.41768 

10.58232 

9.43308 

10.56692 

10.01540 

9-98460 

50 

20 

9.42232 

10.57768 

9.438o6 

10.56194 

10.01574 

9.98426 

40 

30 

9.42690 

io.573io 

9.44299 

10.55701 

10.01609 

9.98391 

30 

40 

9.43I43 

10.56857 

9.44787 

10.55213 

10.01644 

9.98356 

20 

50 

9-43590 

10.56409 

9.45271 

10.54729 

10.01680 

9-98320 

IO 

16 

0 

9-44034 

10.55966 

9-45750 

10.54250 

10.01716 

9.98284 

0 

74 

10 

9-44472 

10.55528 

9.46224 

10.53776 

10.01752 

9.98248 

50 

20 

9.44905 

10.55095 

9-46694 

10.53305 

10.01789 

9.98211 

40 

30 

9-45334 

10.54666 

9.47160 

10.52840 

10.01826 

9.98174 

30 

40 

9-45758 

10.54242 

9.47622 

10.52378 

10.01864 

9-98136 

20 

50 

9.46178 

10.53822 

9.48080 

10.51920 

10.01902 

9.98098 

IO 

17 

o 

9.46594 

10.53406 

9.48534 

10.51466 

10.01940 

9.98060 

o 

73 

IO 

9-47005 

10.52995 

9.48984 

10.51016 

10.01979 

9.98021 

50 

20 

9-474" 

10.52589 

9-49430 

10.50570 

10.02018 

9.97982 

40 

30 

9.47814 

10.52186 

9.49872 

10.50128 

10.02058 

9-97942 

30 

40 

9-48213 

10.51787 

9.50311 

10.49689 

10.02098 

9.97902 

20 

50 

9.48607 

I0.5I393 

9.50746 

10.49254 

10.02138 

9.97861 

10 

18 

o 

9.48998 

10.51002 

9.5H78 

10.48822 

10.02179 

9.97821 

o 

78 

10 

9.49385 

10.50615 

9.51606 

10.48394 

10.02221 

9-97779 

50 

20 

9-49768 

10.50232 

9-52031 

10.47969 

10.02262 

9-97738 

40 

30 

9-50I48 

10.49852 

9.52452 

10.47548 

IO.O23O4 

9.97696 

30 

40 

9.50523 

10.49477 

9.52870 

10.47130 

10.02347 

9.97653 

20 

50 

9.50896 

10.49104 

9.53285 

10.46715 

IO.O2390 

9.97610 

10 

19 

0 

9-5I264 

10.48736 

9.53697 

10.46303 

10.02433 

9-97567 

0 

71 

IO 

9.51629 

10.48371 

9.54io6 

10.45894 

10.02477 

9.97523 

50 

20 

9.5I99I 

10.48009 

9-54512 

10.45488 

10.02521 

9-97479 

40 

30 

9.52350 

10.47650 

9-54915 

10.45085 

10.02565 

9-97435 

30 

40 

9.52705 

10.47295 

9-55315 

10.44685 

10.02610 

9-97390 

20 

50 

9-53057 

10.46944 

9-55712 

10.44288 

10  .  02656 

9-97344 

IO 

20 

0 

9-53405 

10.46595 

9.56107 

10.43893 

10.02701 

9.97299 

o 

70 

10 

9-53751 

10.46249 

9.56498 

10.43502 

10.02748 

9.97252 

50 

20 

9-54093 

10.45907 

9.56887 

10.43113 

10.02794 

9.97206 

40 

30 

9-54433 

10.45567 

9.57274 

10.42726 

10.02841 

9-97159 

30 

40 

9.54769 

10.45231 

9.57658 

10.42342 

10.02889 

9.97111 

20 

50 

9  55102 

10.44898 

9-58039 

10.41961 

10.02936 

9-97063 

10 

69 

Cosine. 

Sec. 

Cotang. 

Tang. 

Cosec. 

Sine. 

For  functions  from  69°  lo7  to  76°  oo'  read  from  bottom  of  table  upward. 


206  PRACTICAL  SURVEYING 

LOGARITHMIC  FUNCTIONS  OF  ANGLES  (Continued) 


Deg. 

Min. 

Sine. 

Cosec. 

Tang. 

Cotang. 

Sec. 

Cosine. 

Min. 

Deg. 

21 

0 

9-55433 

10.44567 

9-58418 

10.41582 

10.02985 

9-97015 

0 

69 

10 

9.5576i 

10.44239 

9.58794 

10.41206 

10.03034 

.9-96966 

50 

20 

9-56085 

io.439i5 

9.59168 

10.40832 

10.03083 

9.96917 

40 

30 

9.56408 

10.43592 

9-59540 

10.40460 

10.03132 

9.96868 

30 

40 

9-56727 

10.43273 

9-59909 

10.40091 

10.03182 

9.96818 

20 

50 

9-57044 

10.42956 

9  .  60276 

10.39724 

10.03233 

9-96767 

10 

22 

0 

9.57358 

10.42642 

9.60641 

10.39359 

10.03283 

9.96717 

o 

68 

10 

9-57669 

10.42331 

9.61004 

10.38996 

10.03335 

9-96665 

50 

20 

9.57978 

10.42022 

9-61364 

10.38636 

10.03386 

9.96614 

40 

30 

9-58284 

10.41716 

9.61722 

10.38278 

10.03438 

9.95562 

30 

40 

9-53583 

10.41412 

9.62079 

10.37921 

10.03491 

9-96509 

20 

50 

9.58889 

10.41111 

9-62433 

10.37567 

10.03544 

9-96456 

10 

23 

0 

9.59188 

10.40812 

9-62785 

10.37215 

10.03597 

9.96403 

0 

67 

10 

9.59484 

10.40516 

9.63135 

10.36365 

10.03651 

9.96349 

50 

20 

9.59778 

10.40222 

9.63484 

10.36516 

10.03706 

9-96294 

40 

30 

9  .  60070 

io.3993o 

9-63830 

10.36170 

10.03760 

9.96240 

30 

40 

9.60359 

10.39641 

9.64175 

10.35825 

10.03815 

9.96185 

20 

50 

9.60646 

10.39354 

9.64517 

10.35483 

10.03871 

9.96129 

10 

24 

0 

9  60931 

10.39069 

9-64858 

10.35142 

10.03927 

9-96073 

0 

C6 

10 

9.61214 

10.38786 

9.65197 

10.34803 

10.03983 

9.96017 

50 

20 

9.61494 

10.38506 

9.65535 

10.34465 

10.04040 

9.9596o 

40 

30 

9.6i773 

10.38227 

9-65870 

10.34130 

10  .  04098 

9-95902 

30 

40 

9-62049 

10.37951 

9.66204 

10.33795 

10.04156 

9.95845 

20 

50 

9-62323 

10.37677 

9.66537 

10.33463 

10.04214 

9.95786 

IO 

25 

0 

9-62595 

10.37405 

9.66867 

10  33133 

10.04272 

9  95728 

0 

66 

10 

9.62865 

10.37135 

9.67196 

10.32804 

10.04332  |  9.95668 

50 

20 

9-63I33 

10.36867 

9-67524 

10.32476 

10.04391 

9.95609 

40 

30 

9.63398 

10.36602 

9.67850 

10.32150 

10.04451 

9-95549 

30 

40 

9.63662 

10.36338 

9.68174 

10.31826 

10.04512 

9.95488 

20 

50 

9.63924 

10.36076 

9-68497 

10.31503 

10.04573 

9.95427 

10 

26 

0 

9.64184 

10.35816 

9.68818 

10.31182 

10.04634 

9.95366 

o 

64 

10 

9-64442 

10.35558 

9.69138 

10.30862 

10.04696 

9-95304 

50 

20 

9.64698 

10.35302 

9-69457 

10.30543 

10.04758 

9.95242 

40 

30 

9.64953 

10.35047 

9.69774 

10.30226 

10.04821 

9.95179 

30 

40 

9-65205 

10.34795 

9.70089 

'10.29911 

10.04884 

9-9SII6 

20 

50 

9.65456 

10.34544 

9.70404 

10.29596 

10.04948 

9-95052 

10 

27 

o 

9.65705 

10.34295 

9.70717 

10  .  29283 

10.05012 

9.94988 

o 

63 

10 

9.65952 

10.34048 

9.71028 

10.28972 

10.05077 

9-94923 

50 

20 

9.66197 

10.33803 

9.71338 

10.28661 

10.05142 

9.94858 

40 

30 

9.66441 

10.33559 

9-71648 

10.28352 

10.05207 

9-94793 

30 

40 

9.66682 

I0.333I8 

9-71955 

10.28045 

10.05273 

9.94727 

20 

50 

9-66923 

10.33078 

9.72262 

10.27738 

10.05340 

9.94660 

10 

62 

Cosine. 

Sec. 

Cotang. 

Tang. 

Cosec. 

Sine. 

For  functions  from  62°  10'  to  69°  oo'  read  from  bottom  of  table  upward. 


TRIGONOMETRY 

LOGARITHMIC  FUNCTIONS  OF  ANGLES  (Continued) 


207 


Deg. 

Min. 

Sine. 

Cosec. 

Tang. 

Cotang. 

Sec. 

Cosine. 

Min. 

Deg. 

28 

o 

9.67161 

10.32839 

9.72567 

10.27433 

10.05406 

9-94593 

o 

62 

10 

9.67398 

10.32602 

9.72872 

10.27128 

10.05474 

9-94526 

50 

20 

9-67633 

10.32367 

9-73175 

10.26825 

10.05542 

9-94458 

40 

30 

9.67866 

10.32134 

9-73476 

10.26524 

10.05610 

9-94390 

30 

40 

9.68098 

10.31902 

9-73777 

10.26223 

10.05679 

9-94321 

20 

SO 

9.68328 

10.31672 

9.74077 

10.25923 

10.05748 

9.94252 

10 

29 

0 

9-68557 

10.31443 

9-74375 

10.25625 

10.05818 

9.94182 

o 

61 

IO 

9.68784 

10.31216 

9.74673 

10.25327 

10.05888 

9.94112 

50 

20 

9.69010 

10.30990 

9.74969 

10.25031 

10.05959 

9  94041 

40 

30 

9-69234 

10.30766 

9.75264 

10.24736 

10.06030 

9-93970 

30 

40 

9.69456 

10.30544 

9-75558 

10.24442 

10.06102 

9.93898 

20 

50 

9-69677 

10.30323 

9.75852 

10.24148 

10.06174 

9-93826 

10 

SO 

0 

9.69897 

10.30103 

9.76144 

10.23856 

10.06247 

9-93753 

0 

60 

IO 

9.70115 

10.29885 

9.76435 

10.23565 

10.06320 

9.9368o 

50 

20 

9-70332 

10.29668 

9.76726 

10.23275 

10.06394 

9.936o6 

40 

30 

9.70547 

10.29453 

9-77015 

10.22985 

10.06468 

9-93532 

30 

40 

9.70761 

10.29239 

9.77303 

10.22697 

10.06543 

9-93457 

20 

So 

9.70973 

10.29027 

9-77591 

10.22409 

10.06618 

9-93382 

10 

31 

0 

9.71184 

10.28816 

9.77877 

10.22123 

10.06693 

9.93307 

o 

59 

10 

9-71393 

10.28607 

9-78163 

10.21837 

10.06770 

9-93230 

50 

20 

9.71602 

10.28398 

9.78448 

10.21552 

10.06846 

9-93154 

40 

30 

9.71809 

10.28191 

9.78732 

10.21268 

10.06923 

9.93077 

30 

40 

9.72014 

10.27986 

9-79015 

10.20985 

10.07001 

9.92999 

20 

50 

9.72218 

10.27782 

9.79297 

10  .  20703 

10.07079 

9.92921 

10 

32 

o 

9.72421 

10.27579 

9-79579 

10.20421 

10.07158 

9.92842 

O 

58 

10 

9.72622 

10.27378 

9.79860 

10.20140 

10.07237 

9.92763 

50 

20 

9.72823 

10.27177 

9.80140 

10.19860 

10.07317 

9.92683 

40 

30 

9-73022 

10.26978 

9.80419 

10.19581 

10.07397 

9.92603 

30 

40 

9.73219 

10.26781 

9-80697 

10.19303 

10.07478 

9.92522 

20 

50 

9.73416 

10.26584 

9.80975 

10.19025 

10.07559 

9.92441 

10 

33 

0 

9.73611 

10.26389 

9.81252 

10.18748 

10.07641 

9  92359 

o 

57 

10 

9.73805 

10.26195 

9.81528 

10.18472 

10.07723 

9.92277 

50 

20 

9-73997 

10.26003 

9.81803 

10.18197 

10.07806 

9.92II94 

40 

30 

9.74189 

10.25811 

9.82078 

10.17922 

10.07889 

9.92111 

30 

40 

9-74379 

10.25621 

9-82352 

10.17648 

10.07973 

9.92027 

20 

50 

9.74568 

10.25432 

9.82626 

10.17374 

10.08058 

9.91942 

IO 

34 

o 

9.74756 

10.25244 

9-82899 

10.17101 

10.08143 

9.91857 

0 

56 

IO 

9-74943 

10.25057 

9-83I7I 

10.16829 

10.08228 

9.91772 

50 

20 

9.75I28 

10.24872 

9.83442 

10.16558 

10.08314 

9.91686 

40 

30 

9-75313 

10.24687 

9.837I3 

10.16287 

10.08401 

9.91599 

30 

40 

9.75496 

10.24504 

9.83984 

10.16016 

10.08488 

9.9ISI2 

20 

50 

9.75678 

10.24322 

9.84254 

10.15746 

10.08575 

9-9I425 

10 

55 

Cosine. 

Sec. 

Cotang. 

Tang. 

Cosec. 

Sine. 

For  functions  from  55°  10'  to  62°  oc/  read  from  bottom  of  table  upward. 


208  PRACTICAL   SURVEYING 

LOGARITHMIC  FUNCTIONS  OF  ANGLES  (Continued) 


Deg. 

Min. 

Sine. 

Cosec. 

Tang. 

Cotang. 

Sec. 

Cosine. 

Min. 

Deg. 

35 

o 

9.75859 

10.24141 

9-84523 

10.15477 

10.08664 

9.91336 

o 

55 

10 

9-76039 

10.23961 

9.84791 

10.15209 

10.08752 

9-91248 

50 

20 

9.76218 

10.23782 

9-85059 

10.14941 

10.08842 

9.9H58 

40 

30 

9.76395 

10.23604 

9-85327 

10.14673 

10.08931 

9-91069 

30 

40 

9.76572 

10.23428 

9.85594 

10.14406 

10.09022 

9-90978 

20 

50 

9.76747 

10.23253 

9.85860 

10.14140 

10.09113 

9-90887 

IO 

36 

o 

9.76922 

10.23078 

9.86126 

10.13874 

10.09204 

9.90796 

0 

54 

10 

9.77095 

10.22905 

9-86392 

10.13608 

10  .  09296 

9.90704 

50 

20 

9.77268 

10.22732 

9.86656 

10.13344 

10.09389 

9.90611 

40 

30 

9-77439 

10.22561 

9.86921 

10.13079 

10.09482 

9-90518 

30 

40 

9.77609 

10.22391 

9-87185 

10.12815 

10.09576 

9.90424 

20 

50 

9-77778 

I  O.  22222 

9.87448 

10.12552 

10.09670 

9.90330 

10 

37 

o 

9.77946 

IO.22O54 

9-87711 

10.12289 

10.09765 

9.90235 

0 

53 

10 

9.78H3 

IO.2I887 

9.87974 

10.12026 

10.09861 

9-90139 

50 

20 

9.78280 

I0.2I72O 

9.88236 

10.11764 

10.09957 

9-90043 

40 

30 

9.78445 

10.21555 

9.88498 

10.11502 

10.10053 

9.89947 

30 

40 

9-78609 

10.21391 

9.88759 

10.11241 

10.10151 

9.89849 

20 

50 

9.78772 

10.21228 

9.89020 

10.10980 

10.10248 

9-89752 

10 

38 

o 

9.78934 

IO.2IO66 

9.89281 

10.10719 

10.10347 

9.89653 

o 

52 

IO 

9-79095 

10.20905 

9.89541 

10.10459 

10.10446 

9.89554 

So 

20 

9.79256 

10.20744 

9.89801 

10.10199 

10.10545 

9.89455 

40 

30 

9-79415 

10.20585 

9.90061 

10.09939 

10.10646 

9.89354 

30 

40 

9-79573 

10.20427 

9.90320 

10.09680 

10.10746 

9.89254 

20 

50 

9-79731 

10.20269 

9.90578 

10.09422 

10.10848 

9.89152 

10 

39 

o 

9.79887 

IO.2OII3 

9.90837 

10.09163 

10.10950 

9-89050 

o 

51 

10 

9-80043 

10.19957 

9-91095 

10.08905 

10.11052 

9.88948 

50 

20 

9.80197 

10.19803 

9.91353 

10.08647 

10.11156 

9.88844 

40 

30 

9-80351 

10.19649 

9.91610 

10.08390 

10.11259 

9.88741 

30 

40 

9-80504 

IO.I9496 

9.91868 

10.08132 

10.11364 

9.88636 

20 

50 

9.80656 

10.19344 

9.92125 

10.07875 

10.11469 

9-88531 

10 

40 

o 

9.80807 

10.19193 

9.92381 

10.07619 

10.11575 

9.88425 

O 

50 

TO 

9.80957 

IO.I9O43 

9-92638 

10.07362 

10.  11681 

9.88319 

50 

2O 

9.81106 

10.18894 

9.92894 

10.07106 

10.11788 

9.88212 

40 

30 

9-81254 

10.18746 

9.93150 

10.06850 

10.11895 

9.88105 

30 

40 

9.81402 

10.18598 

9.93406 

10.06594 

10.12004 

9.87996 

20 

50 

9-81548 

IO.I845I 

9.93661 

10.06339 

10.12113 

9.87887 

IO 

41 

0 

9.81694 

10.18306 

9.939i6 

10.06084 

10.12222 

9.87778 

o 

49 

10 

9  81839 

10.  18161 

9.94171 

10.05829 

10.12332 

9.87668 

50 

20 

9  81983 

10.18017 

9.94426 

10.05574 

10.12443 

9-87557 

40 

30 

9.82126 

10.17874 

9.94681 

10.05319 

10.12554 

9.87446 

30 

40 

9.82269 

10.17731 

9-94935 

10.05065 

10.12666 

9-87334 

?o 

50 

9.82410 

10.17590 

9.95190 

10.04810 

10.12779 

9.87221 

10 

48 

Cosine. 

Sec. 

Cotang. 

Tang. 

Cosec. 

Sine. 

For  functions  from  48°  10'  to  55°  oo'  read  from  bottom  of  table  upward 


TRIGONOMETRY 
LOGARITHMIC  FUNCTIONS  OF  ANGLES  (Continued) 


209 


1 

Deg. 

Min. 

Sine. 

Cosec. 

Tang. 

Cotang. 

Sec. 

Cosine. 

Min. 

Deg. 

42 

o 

9-82551 

10.17449 

9-95444 

10.04556 

10.12893 

9.87107 

o 

48 

10 

9.82691 

10.17309 

9.95698 

10.04302 

10.13007 

9.86993 

So 

20 

9.82830 

10.17170 

9-95952 

10.04048 

10.13121 

9.86879 

40 

30 

9.82968 

10.17032 

9.96205 

10.03795 

10.13237 

9.86763 

30 

40 

9.83106 

10.16894 

9.96459 

10.03541 

10.13353 

9.86647 

20 

So 

9.83242 

10.16758 

9.96712 

10.03288 

10.13470 

9.86530 

10 

43 

0 

9.83378 

10.16622 

9.96966 

10.03034 

10.13587 

9-86413 

o 

47 

10 

9-83513 

10.16487 

9.97219 

10.02781 

10.13705 

9-86295 

So 

20 

9.83648 

10.16352 

9-97472 

10.02528 

10.13824 

9.86176 

40 

30 

9.83781 

10.16219 

9-97725 

10.02275 

10.13944 

9.86056 

30 

40 

9.83914 

10.16086 

9.97978 

10.02022 

10.14064 

9  85936 

20 

SO 

9.84046 

10.15954 

9.98231 

IO.OI7O9 

10.14185 

9-85815 

IO 

44 

o 

9.84177 

10.15823 

9-98484 

IO.OI5I6 

10.14307 

9-85693 

O 

46 

10 

9.84308 

10.15692 

9.98737 

10.01263 

10.14429 

9.85571 

So 

20 

9-84437 

10.15563 

9.98989 

IO.OIOII 

10.14552 

9.85448 

40 

30 

9.84566 

10.15434 

9.99242 

10.00758 

10.14676 

9-85324 

30 

40 

9-84694 

10.15306 

9-99495 

IO.OO5O5 

10.14800 

9.85200 

20 

So 

9.84822 

10.15178 

9-99747 

IO.O0253 

10.14926 

9-85074 

10 

46 

9.84949 

10.15052 

10.00000 

10.00000 

IO  .  15052 

9.84949 

46 

Cosine. 

Sec. 

Cotang. 

Tang. 

Cosec. 

Sine. 

For  functions  from  45°  09' to  48°  oo'  read  from  bottom  of  table  upward. 


CHAPTER   VI 

TRANSIT   SURVEYING 

Surveying  was  an  established  calling  many  centuries  be- 
fore the  compass  was  known,  there  being  a  well-developed 
system  of  mensuration  in  Egypt  in  the  time  of  Joseph. 
Nothing  however  is  known  of  the  instruments  used  in  the 
most  ancient  times. 

In  the  second  century  B.C.  there  lived  in  Alexandria 
Heron  the  Elder,  a  mathematician  and  practical  surveyor. 
He  has  been  styled  "The  first  engineer,"  because  of  a 
number  of  inventions,  one  being  the  aeolipile,  the  first 
steam  engine.  A  book  entitled  "Dioptra,"  the  first 
known  treatise  on  surveying,  is  supposed  to  have  been 
written  by  him  although  some  writers  believe  it  to  be  the 
work  of  a  later  writer  of  the  same  name.  The  "Dioptra" 
is  a  treatise  on  the  use  of  the  diopter,  a  surveying  instru- 
ment in  common  use  up  to  the  end  of  the  Middle  Ages. 
With  this  instrument  the  Romans  laid  out  their  cities, 
roads,  aqueducts  and  all  public  works. 

Venturi  wrote :  "  Dioptra  were  instruments  resembling  the 
modern  theodolites.  The  instrument  consisted  of  a  rod, 
four  yards  long,  with  little  plates  at  the  end  for  aiming. 
This  rested  upon  a  circular  disk.  The  rod  could  be  moved 
horizontally  and  also  vertically.  By  turning  the  rod 
around  until  stopped  by  two  suitably  located  pins  on  the 
circular  disk,  the  surveyor  could  work  off  a  line  perpendic- 
ular to  a  given  direction.  The  level  and  plumb  line  were 
also  used."  From  an  illustration  given  by  Heron  of  a 
simple  diopter  and  from  an  illustration  given  by  Venturi  of 
a  later  type  it  is  evident  that  the  instrument  was  merely  a 
large  surveyors'  cross  for  setting  out  perpendiculars.  At 
what  date  the  plate  was  graduated  so  other  angles  could 
be  set  off  no  one  seems  to  know.  The  compass  was  not 
long  in  use  before  a  needle  was  placed  on  the  plate  of  the 

210 


TRANSIT  SURVEYING 


211 


diopter,  later  followed  by  two  concentric  plates  so  that  angles 
could  be  read  independently  of  the  needle.  Transversals 
were  used  to  subdivide  the  graduations  to  enable  fine 
readings  to  be  taken. 

About  the  year  1608  a  Dutch  spectacle  maker  named 
Lippershey  discovered  the  principle  of  the  telescope  and 
Galileo,  in  1609,  made  the  first  telescope.  In  1631  the 
vernier  was  invented  and  in  1640  cross-hairs  were  used  to 
define  the  optical  axis,  or  line  of  sight,  of  a  telescope. 
When  the  vernier  was  used  to  read  the  graduated  circles 
and  the  improved  telescope 
took  the  place  of  the  sighting 
disks,  the  diopter  became  a 
theodolite.  This  instrument 
seems  to  have  been  first  men- 
tioned in  print  about  the  year 
1674.  The  first  English  tele- 
scopic theodolite  is  believed 
to  have  been  made  in  1723. 
Fig.  1 80,  copied  from  an  old 
edition  of  Davies'  Survey- 
ing, shows  the  type  in  use 
in  1835.  The  etymology  of 
the  word  is  doubtful,  some 
writers  believing  it  to  come 
from  three  Greek  words  mean- 
ing "to  see  a  way  plainly," 
while  others  believe  it  to  have 
been  named  as  a  compliment 
to  M.  Theodolus,  a  French 
mathematician  who  wrote  a  treatise  concerning  its  use,  and 
who  may  have  been  the  man  who  was  responsible  for  the 
final  form  the  diopter  assumed  before  the  name  was  changed. 
The  first  may  have  been  called  "Dioptra  Theodolus." 

The  theodolite  was  not  well  adapted  to  the  work  re- 
quired of  a  surveying  instrument  in  America  and  the 
compass  held  sway  until  the  commencement  of  steam  rail- 
way construction.  The  graduated  plates  with  verniers 
made  the  theodolite  a  good  instrument  for  laying  out  rail- 
way curves,  a  work  for  which  the  compass  was  plainly  riot 
fit.  The  telescope  with  its  cross-hairs  enabled  points  to 


FIG.  1 80.     Cradle  theodolite. 


212  PRACTICAL  SURVEYING 

be  set  with  an  accuracy  equal  to  that  of  the  graduations 
but  rapid  work  could  not  be  done  because  the  telescope 
was  mounted  in  wyes  (cradles)  and  could  only  be  reversed 
for  backsights  by  changing  it  in  the  wyes  end  for  end. 
In  1831  Mr.  Young,  an  instrument  maker  in  Philadelphia, 
applied  the  principle  of  the  astronomical  transit  instru- 
ment to  a  mounting  for  a  theodolite  telescope  and  the 
"Portable  American  Transit  for  Engineers"  appeared.  It 
is  the  standard  surveying  instrument  in  America  today 
for  everything  but  work  of  the  highest  character.  On 
such  work  theodolites  are  used,  the  word  in  the  United 
States  being  limited  to  high-grade  surveying  instruments 
without  a  compass,  the  omission  of  the  needle  permitting 
of  great  rigidity  in  the  mounting  of  the  telescope.  Few,  if 
any,  "cradle"  theodolites  are  made.  In  Europe  the  word 
theodolite  is  used  to  describe  all  surveying  instruments 
having  graduated  horizontal  and  vertical  circles,  with  and 
without  needles,  and  regardless  of  how  the  telescopes  are 
mounted. 

The  transit  is  mounted  on  a  vertical  compound  center. 
That  is,  the  vertical  shaft  has  a  central  solid  spindle  fitting 
in  an  outer  one  which  works  in  a  deep  socket  attached  to 
the  leveling  device.  The  outer  portion  of  the  vertical 
center  carries  a  horizontal  disk  with  a  circumferential  band 
of  silver  on  top,  the  inner  edge  of  the  band  being  graduated. 
The  solid  spindle  carries  a  horizontal  disk  extending  over 
the  graduated  ring.  Upon  this  covering  plate  is  placed  the 
standards  carrying  the  telescope,  verniers  for  reading  the 
graduations  closely  and  a  compass. 

The  compass  needle  is  as  long  as  possible,  for  much  work 
is  done  with  the  needle  in  sections  of  the  country  .where 
land  is  cheap.  The  compass  is  always  fitted  with  a  screw 
for  lifting  the  needle  and  is  usually  supplied  with  a  varia- 
tion plate.  The  compass  box  occupies  much  space  and  it 
is  necessary  to  place  the  telescope  standards  near  the  edge 
of  the  plate.  This  lessens  the  stability  somewhat,  in  the 
opinion  of  hypercritical  persons,  but  not  enough  to  affect 
much  of  the  work  done  by  the  majority  of  engineers  and 
surveyors.  For  high-grade  city  surveying  the  standards 
are  U-shape,  thus  bringing  them  closer  to  the  center  at 
the  base,  and  a  trough  compass  is  used.  The  trough  com- 


TRANSIT  SURVEYING 


213 


pass  consists  of  a  needle  mounted  in  a  narrow  box  so  it 
can  swing  only  a  few  degrees  to  the  right  or  left  of  the 
meridian.  A  needle  thus  mounted  serves  to  set  the  line  of 
sight  in  the  magnetic  meridian,  from  which  base  line  other 
courses  are  run  wholly  by  angles  read  on  the  horizontal 
plate. 

In  Fig.  181  is  shown  a  typical  American  transit  used  for 
most  of  the  work  done  by  engineers  and  surveyors.  The 
transit  is  attached  to  a  tripod, 
the  upper  end  of  which  is  shown. 

A.  The   ring   containing    the 
threads  for  attaching  the  transit 
to  the  tripod.     It  is  often  called 
the  "lower  screw  plate." 

B.  The  leveling  screws.  These 
screws  work  in  dustproof  screw 
caps  attached  to  a  plate,  made 
in  the  shape  of  a  cross  for  light- 
ness.    This  cross,  "upper  screw 
plate,"  contains    the   socket   in 
which     the     instrument    center 
spindle  works.      On    the   lower 
end  of  this  socket  is  attached  a 
ball   joint   working    in    a   plate 
underneath     the     lower     screw 
plate.     The  hole  in  this  plate  is 
much    larger    than    the    socket 

stem  so  by  loosening  the  screws  the  entire  instrument 
may  be  moved  from  one  side  to  the  other,  the  device  being 
termed  a  "shifting  center." 

Four  screws  are  commonly  used,  but  for  many  years 
only  three  screws  have  been  customary  on  the  highest 
grade  instruments  for  geodetic  work.  Within  recent  years 
three-screw  bases  have  been  made  which  are  almost  as 
compact  as  four-screw  bases,  and  many  engineers  now 
have  three  instead  of  four  leveling  screws  on  their  transits 
and  levels.  No  man  who  is  accustomed  to  using  three 
screws  goes  back  willingly  to  four.  The  latter  are  much 
the  slower,  are  not  as  stable  and  require  the  use  of  two 
hands  to  keep  the  bubbles  centered  in  the  plate  levels. 
Leveling  screws  are  used  to  keep  the  horizontal  plate 


FIG.  181. 


Modern  American 
transit. 


214  PRACTICAL   SURVEYING 

level.     If  the  plate  is  not  level  all  horizontal  angles  read 
will  be  too  small. 

C.  The   lower   clamp   screw.     This   screw   works   in   a 
clamp  attached  to  the  spindle  socket,  the  inner  end  of  the 
screw  resting  against  a  block.     When  turned  it  presses  the 
block  against  the  outer  spindle  thus  clamping  the  spindle 
to  the  socket. 

D.  Lower  tangent  screw.     This  is  fastened  to  the  upper 
screw  plate  and  works  against  the  lower  clamp  screw  for  the 
purpose  of  making  a  final  adjustment.     Formerly  two  op- 
posing screws  were  used  but  now  there  is  but  one  screw  as 
shown,  the  cylindrical  case  on  the  other  side  of  the  clamp 
screw  containing  a  strong  German  silver  spring. 

E.  The  upper  clamp  screw  used  to  clamp  the  inner  and 
outer  spindles  together.. 

F.  The  upper  tangent  screw  with  opposing  spring  for 
making  a  final  adjustment  of  the  line  of  sight. 

G.  The   needle   lifter.     A   similar   screw   not   shown   is 
used  to  shift  the  variation  plate. 

H.  Clamp  screw  for  the  vertical  arc.  This  screw  is  on 
an  arm  which  it  clamps  to  the  axis  of  the  telescope.  The 
lower  end  of  the  arm  is  between  the  vertical  arc  tangent 
screw  and  opposing  spring. 

/.  Vertical  arc  tangent  screw.  After  the  clamp  screw 
clamps  the  telescope  axis  the  tangent  screw  is  used  to  make 
the  final  small  vertical  movement  required  for  accurate 
pointing. 

The  graduated  disk  shown  ahead  of  the  vertical  arc 
tangent  screw  is  a  gradienter.  The  author  had  two  tran- 
sits equipped  with  gradienters  and  used  these  transits  for 
more  than  sixteen  years.  In  all  that  time  he  found  no 
occasion  when  he  could  use  a  gradienter  to  advantage. 
Furthermore  he  has  yet  to  meet  an  engineer  who  uses  a 
gradienter  in  preference  to  stadia  wires  or  the  vertical  arc. 
In  purchasing  a  transit  five  dollars  are  saved  when  the 
gradienter  is  omitted.  If  the  gradienter  is  to  be  of  service 
the  threads  of  the  tangent  screw  must  be  accurately  cut  and 
be  in  perfect  condition.  After  a  few  years  of  use  all  screw 
threads  are  worn  and  while  slight  wear  does  not  affect  them 
for  use  on  tangent  screws  it  does  make  them  unfit  for  mi- 
crometer work,  the  gradienter  being  a  form  of  micrometer. 


TRANSIT  SURVEYING  215 

Two  small  levels  are  used  to  level  the  horizontal  plate. 
One  is  set  parallel  with  the  line  of  sight,  usually  on  the  left 
standards  as  shown  in  the  cut.  The  other  is  set  across  the 
line  of  sight  on  the  edge  of  the  plate  under  the  object  end 
of  the  telescope,  and  is  the  more  important  of  the  two. 
The  bubbles  are  centered  by  means  of  the  leveling  screws  B. 

Through  a  rectangular  opening  in  the  upper  plate  the 
graduations  and  vernier  are  viewed.  This  opening  is 
covered  with  glass  and  a  reflector  of  celluloid,  or  ground 
glass,  is  used  to  enable  the  graduations  to  be  seen  readily. 

On  the  best  transits  double  verniers  are  used,  180  de- 
grees apart,  so  readings  may  be  checked  and  to  assist  in 
repeating  readings.  Some  makers  place  the  vernier  on  the 
sides  between  the  standards;  others  directly  under  the  tele- 
scope, but  the  best  place  is  30  degrees  to  the  left  of  the 
line  of  sight  and  is  most  common.  In  this  position  the 
vernier  may  be  read  without  disturbing  the  telescope  and 
the  surveyor  does  not  have  to  step  around  the  instrument. 

The  telescope  is  mounted  on  an  axis  revolving  in  bear- 
ings on  top  of  the  standards.  It  is  generally  of  such  a 
length  as  to  permit  of  a  complete  revolution,  but  with  a 
sunshade  on  the  object  end  the  eye  end  only  will  clear  the 
compass  glass.  The  small  capstan  screws  on  the  telescope 
tube  indicate  the  location  of  the  cross-wires,  which  are 
brought  into  the  field  of  view  by  moving  the  eyepiece  in 
or  out.  This  focusing  is  done  in  one  of  several  ways:  by 
a  straight  pull,  by  a  screwing  motion  or  by  means  of  a 
screw  near  the  eyepiece  which  moves  a  rack  and  pinion 
within  the  tube.  The  object  glass  is  focused  by  means 
of  a  screw  on  the  right-hand  side  of  the  telescope  and  is 
therefore  not  shown  in  the  cut.  Some  makers  place  this 
screw  on  top. 

For  taking  vertical  angles  a  vernier  is  attached  to  the 
left-hand  standards  and  an  arc  or  a  full  circle  is  attached  to 
the  telescope  axis.  For  practically  ninety-five  per  cent  of 
work  done  by  surveyors  an  arc  like  that  shown  in  the  cut 
is  sufficient  and  the  author  prefers  it  to  a  full  circle  which 
is  more  apt  to  be  injured  because  of  its  projection  above 
the  top  of  the  standards.  A  full  circle  is  particularly  ex- 
posed to  damage  in  brushy  land.  When  long  lines  are 
run  and  elevations  are  taken  by  means  of  vertical  angles, 


21 6  PRACTICAL  SURVEYING 

instead  of  a  regular  level,  as  happens  frequently  in  under- 
ground surveying,  a  full  circle  with  double  and  opposite 
verniers  is  advisable.  It  should  be  enclosed  in  a  protective 
shield  with  glazed  verniers. 

Under  the  telescope  is  shown  a  long  level.  When  in  ad- 
justment this  may  be  used  for  leveling,  the  transit  being 
therefore  a  leveling  instrument  as  well  as  an  angle  measurer. 
First  level  the  instrument  so  the  bubbles  in  the  plate 
levels  remain  stationary  during  a  complete  revolution 
horizontally.  Then  place  the  telescope  in  as  nearly  a 
horizontal  position  as  possible  and  clamp  the  axis.  By 
means  of  the  vertical  tangent  screw  bring  the  bubble  of 
the  long  level  under  the  telescope  to  the  center.  If  in  ad- 
justment it  will  remain  stationary  during  a  revolution  of 
the  instrument  on  the  vertical  axis.  If  left  in  this  position 
it  will  not  be  necessary  on  succeeding  "set-ups"  to  first 
level  the  plate,  the  telescope  being  alternately  leveled  over 
each  pair  of  screws,  precisely  like  an  engineer's  level. 
Accurate  leveling  may  be  done  with  a  transit  but  it  is 
slow  compared  with  a  level. 

When  the  transit  is  not  being  used  as  a  level  the  telescope 
should  be  pointed  vertically,  or  in  line  with  the  vertical 
axis  when  the  instrument  is  carried  between  stations.  It 
should  be  clamped  as  lightly  as  possible.  In  this  way  it 
offers  the  least  obstruction  in  brush  and  a  blow  will  cause 
it  to  revolve  on  the  axis.  The  lower  tangent  screw  should 
be  loose,  leaving  the  instrument  free  to  revolve  if  acci- 
dently  struck. 

A  plain  transit  has  no  level  under  the  telescope  and  no 
vertical  arc  or  circle.  Instrument  makers  quote  on  plain 
transits,  and  give  prices  for  extras.  Plain  transits  are  used 
on  railway  surveys.  If  the  surveyor  expects  to  do  little 
compass  work  a  large  transit  is  not  necessary  if  of  a  good 
make.  When  the  author  was  young  he  bought  a  transit 
with  a  5-in.  needle.  The  weight  was  18  Ibs.  and  the  tripod 
weighed  10  Ibs.  Although  similar  transits  are  now  made 
the  majority  of  engineers  prefer  transits  weighing  not  to 
exceed  13  Ibs.  with  7-lb.  tripods.  For  general  use  the 
surveyor  will  obtain  very  satisfactory  results  with  a  transit 
having  a  3i-in.  needle,  the  combined  weight  of  transit 
and  tripod  being  under  16  Ibs.  The  writer  for  a  number 


TRANSIT  SURVEYING  217 

of  years  held  a  commission  as  a  U.  S.  Deputy  Mineral 
Surveyor  in  the  far  west,  and  for  mountain  work  had  a 
transit  made  to  order.  The  needle  was  about  2\  ins.  long 
and  the  weight  of  the  transit  was  4^  Ibs.  It  was  carried 
in  a  knapsack  held  by  straps  over  the  shoulders.  The 
folding  tripod  weighed  3^  Ibs.,  and  was.  carried  in  a  small 
case  slung  to  the  bottom  of  the  knapsack.  Intended  for 
mountain  work  only  the  instrument  was  gradually  intro- 
duced on  other  work  until  finally  it  was  used  on  all  the 
work  done  by  the  author  for  nearly  ten  years.  Because 
of  its  lightness  it  suffered  less  damage  from  falls  than 
heavier  transits.  For  the  same  reason  it  exhibited  great 
steadiness  in  a  wind.  This  instrument  of  course  repre- 
sented one  extreme  as  the  first  one  purchased  by  the 
author  represented  another  extreme. 

A  man  with  thick  thumbs  and  large  fingers  cannot  be 
made  to  believe  the  smallest-sized  transit  is  a  reliable  in- 
strument. Lightness  and  portability  are  greatly  appre- 
ciated in  rough  country  as  also  in  high  altitudes  where 
breathing  is  a  conscious  effort.  In  lower  country  more 
weight  is  not  objectionable  when  it  implies  more  easily 
read  graduations.  For  the  highest  grade  of  engineering 
surveying,  such  as  re-tracing  boundaries  of  expensive  city 
property,  setting  out  lines  for  great  bridges,  important 
tunnels,  subways,  etc.,  the  need  for  absolute  accuracy 
makes  the  question  of  weight  of  relatively  small  importance. 
With  such  instruments,  however,  a  compass  is  not  required 
and  this  fact  effects  a  saving  in  weight  by  a  gain  in  com- 
pactness. 

Graduations  were  formerly  made  on  silver-coated  brass 
plates.  Now  they  are  cut  in  the  surface  of  a  silver  ring 
attached  to  the  plate.  All  of  the  best  makers  today  cut 
the  graduations  in  solid  silver,  for  by  this  means  only  can 
proper  results  be  obtained. 

For  the  horizontal  circle  five  methods  are  used  for  num- 
bering the  graduations. 

/.  Figured  clockwise  (in  the  direction  followed  by  the  hands 
of  a  clock)  from  o°  to  360°  with  single  opposite  verniers. 

//.  Figured  in  quadrants  with  double  opposite  verniers. 
These  quadrants  correspond  to  the  compass  quadrants  so 
angles  may  be  readily  checked  by  the  needle. 


2i8  PRACTICAL  SURVEYING 

///.  A  combination  of  I  and  II  with  double  opposite 
verniers. 

IV.  Figured  in  one  row,  clockwise  and  anti-clockwise, 
from  o°  to  1 80°  each  way  with  double  opposite  verniers. 

V.  Figured  in  two  rows,  one  clockwise  and  one  anti- 
clockwise, from  o°  to  360°,  with  double  opposite  verniers. 
Some  makers  incline  the  figures  in  the  direction  in  which 
they  are  to  be  read  and  some  make  further  provision  against 
error  by  coloring  one  set  red  and  the  other  black. 

Method  V  is  the  best  for  all  purposes  and  the  one  used 
by  instrument  makers  when  the  purchaser  expresses  no 
choice.  Vertical  circles  and  arcs  are  best  graduated 
quadran  tally. 

VERNIERS 

Horizontal  and  vertical  circles  are  divided  into  degrees 
by  short  marks.  Each  fifth  degree  mark  projects  a  little 
and  the  tenth  degree  marks  project  still  more,  the  gradu- 
ations being  figured  at  each  tenth  degree  only.  On  some 
circles  the  degree  is  further  divided  into  quarter-degree, 
third-degree  or  half-degree  graduations.  For  closer  read- 
ings a  vernier  must  be  employed. 

The  vernier  is  a  uniformly  divided  scale  on  a  circle  ad- 
jacent to  the  graduated  circle,  or  limb.  The  end  marks 
,  on  the  vernier  coincide  with  marks  on  the  limb  but  in  the 
included  space  there  is  one  more  interval  on  the  vernier 
than  on  the  limb.  If  we  assume  a  limb  graduated  to  half 
degrees  (30  minutes)  and  29  divisions  on  the  limb  equal 
30  divisions  on  the  vernier  then  each  division  on  the  ver- 
nier is  3*0  smaller  than  a  division  on  the  limb  and  the 
vernier  reads  to  minutes. 

Rule  to  find  reading  of  a  vernier.  Divide  the  least  reading 
of  the  limb  by  number  of  spaces  on  the  vernier. 

Rule  for  reading  an  angle  by-  nteans^  of  a  vernier.  Read 
the  angle  on  the  limlLtp  the 'graduation  nearest  tfie"zero  of  the 
vernier.  Then  read  along  the  vernier  in  the  same  direction 
until  a  line]  is  found  that  coincides  with  some  line  on  the 
limb.  Add  the  vernier  reading  to  the  limb  reading. 

The  following  illustrations  and  descriptions  will  serve  to 
make  the  matter  clear.  The  student  is  advised  to  study 
the  readings  obtained. 


TRANSIT  SURVEYING 


219 


Fig. 

Reading  of 
limb. 

Divisions!      f  Divisions 
of  limb.  J  ~~  lof  vernier 

Reading  of 
vernier. 

Kind,  of  vernier. 

182 

Degrees 

II 

12 

5  minutes 

Double  direct 

183 

30  minutes 

29 

30 

i  minute 

Double  direct 

184 

20  minutes 

39 

40, 

30  seconds 

Double  direct 

i»S 

20  minutes 

59 

60 

20  seconds 

Folded 

186 

30  minutes 

29 

30 

i  minute 

Folded 

187 

15  minutes 

44 

45 

20  seconds 

Double  direct 

188 

15  minutes 

49 

50 

yfa  degree 

Double  direct 

Fig.  182  shows  a  double  direct  vernier,  that  is,  one 
which  reads  from  the  center  to  either  extreme  division  (60), 
that  part  being  used  in  which 
the  direction  of  the  number- 
ing corresponds  to  the  direc- 
tion in  which  the  limb  is  num- 
bered and  read.  The  limb 
is  graduated  to  degrees  and 
the  vernier  (from  o  to  60)  comprises  12  divisions,  there- 
fore the  reading  of  the  vernier  is  60  minutes  -r-  12  =  5 
minutes. 

The  figure  reads  3°  +  50'  =  3°  50'  from  right  to  left. 

Fig.    183   represents   the   usual   graduations   of   an   en- 
gineer's   transit   with   its   vernier.      This   is   an   ordinary 


FIG.  182. 


double-direct  vernier  reading  from  the  center  to  either 
extreme  division  (30).  The  limb  is  graduated  to  half  de- 
grees and  the  vernier  (from  o  to  30)  comprises  30  divisions, 
therefore  the  reading  of  the  vernier  is  30  minutes  -f-  30  =  I 
minute. 

The  figure  reads  27°  +  25'  =  27°  25'  from  left  to  right, 
and  152°  30'  +  05'  =  152°  35'  from  right  to  left. 


220 


PRACTICAL  SURVEYING 


Fig.  184  represents  the  graduations  and  vernier  of  a 
transit  having  finer  graduations  than  that  shown  in  Fig. 
183,  with  a  double-direct  vernier  reading  from  the  center 
to  either  extreme  division  (20).  The  limb  is  graduated  to 


20  minutes  and  there  are  40  divisions  in  the  vernier,  con- 
sequently the  reading  of  the  vernier  is  20  minutes  -f-  40  = 
J  minute  =  30  seconds. 

The  figure  reads  17°  40'  +  12'  30"  =  17°  52'  30"  from 
left  to  right,  and  162°  +  f  30"  =  162°  f  30"  from  right 
to  left. 

In  Fig.  185  the  transit  has  still  finer  divisions  than  those 
already  considered.  The  vernier  is  a  folded  one  reading 
from  the  center,  indicated  by  the  arrow,  to  either  of  the 


extreme  divisions  (10),  and  then  forward  in  the  same 
direction  from  the  other  extreme  division  (10)  to  the 
center  division  (20),  the  direction  being  determined  by  the 
numbering  and  reading  of  the  limb.  The  limb  is  graduated 
to  20  minutes,  while  the  vernier  is  composed  of  60  equal 
parts,  consequently  the  reading  of  the  vernier  is  20  minutes 
-j-  60  =  J  minute  =  20  seconds. 

The  figure  reads  49°  +  14'  20"  =  49°  14'  20"  from  left 
to  right,  and  130°  40'  +  5'  40"  =  130°  45'  40"  from  right 
to  left. 


TRANSIT  SURVEYING 


221 


A  transit  still  more  finely  graduated  is  shown  in  Fig. 
1 86.  This  has  a  double-direct  vernier  reading  from  the 
center  to  either  extreme  division  (45).  The  limb  is  grad- 
uated to  15  minutes  and  there  are  45  divisions  in  the 
vernier,  consequently  the  reading  of  the  vernier  is  15 
minutes  -f-  45  =  ^  minute  =  20  seconds. 

The  figure  reads  30°  +  4'  20"  =  30°  4'  20"  from  left  to 
right  and  149°  45'  +  10'  40"  =  149°  55'  40"  from  right  to 
left. 


FIG.  187. 


Fig.  187  represents  a  portion  of  a  vertical  circle  or  arc 
with  folded  vernier.  The  graduations  are  to  half  degrees, 
and  the  vernier  is  divided  into  30  equal  parts,  consequently 
the  reading  of  the  vernier  is  30  minutes  -5-  30  =  I  minute. 

The  figure  reads  7°  30'  +  21'  =  7°  51'  from  right  to 
left,  an  angle  of  depression. 

In  many  kinds  of  work  a  decimal  vernier  has  some 
minor  advantages  to  recommend  it  and  it  is  extremely 
useful  in  laying  out  railway  curves.  Its  use  is  not  common 
and  many  surveyors  and  engineers  have  never  seen  one. 
Fig.  1 88  shows  the  method  of  graduating  the  horizontal 
limb  and  vernier  to  read  to  decimals  of  a  degree.  This 
vernier  is  a  double-direct 
vernier  reading  from  the 
center  to  either  extreme  di- 
vision (25),  that  part  being 
used  on  which  the  direc- 
tion of  the  numbering  cor- 
responds to  the  direction  in  which  the  limb  is  numbered 
and  read.  The  limb  is  graduated  to  J  degree  (0.25°) 
and  the  vernier  divided  into  50  parts,  consequently  the 
reading  of  the  vernier  is  0.25  -5-  50  =  0.005°  which  equals 
sU  of  a  degree. 

The  figure  reads  45°  +  0.055  =  45-°55°  from  Wt  to 
right  and  314.75°  +  0.195  =  3- 14945°  from  rignt  to 


FIG.  188. 


222  PRACTICAL  SURVEYING 

In  reading  a  vernier  when  the  limb  has  two  rows  of 
figures  care  must  be  used  to  avoid  reading  the  wrong 
row. 

Never  read  the  vernier  by  looking  only  at  one  line. 
The  adjacent  lines  on  each  side  of  the  coinciding  line 
should  be  observed  to  see  that  they  fail  to  coincide  by 
equal  amounts  with  lines  on  the  limb. 

Sometimes  no  coinciding  lines  can  be  found  but  two 
adjacent  lines  will  be  found  which  fail  by  equal  amounts 
to  coincide  with  lines  on  the  limb.  The  true  reading  is 
between  these  lines;  thus  a  closer  reading  may  be  ob- 
tained than  is  indicated  by  the  least  count  of  the  vernier. 

Every  transit  supplied  with  a  vertical  circle  or  arc 
should  be  equipped  with  stadia  wires,  the  use  of  which  will 
be  described  in  the  next  chapter. 

TO   USE  THE  TRANSIT 

The  instrument  should  be  set  up  firmly,  the  tripod  legs 
being  pressed  into  the  ground,  so  as  to  bring  the  plates  as 
nearly  level  as  convenient.  The  plates  should  then  be 
carefully  leveled  and  properly  clamped. 

For  precise  work,  in  addition  to  leveling  by  the  plate 
levels,  it  is  always  advisable,  if  the  transit  has  such  at- 
tachment, to  level  the  plates  by  the  telescope  level,  as  this 
is  much  more  sensitive  than  the  levels  on  the  plate.  In 
this  operation  the  position  of  the  level  on  telescope  must 
be  observed  over  each  pair  of  leveling  screws  in  turn,  and 
one-half  the  correction  made  by  the  axis  tangent  screw, 
the  other  half  by  the  leveling  screws. 

Before  an  observation  is  made  with  the  telescope,  the 
eyepiece  should  be  focused  until  the  object  is  seen  clear 
and  well-defined,  and  the  wires  appear  as  if  fastened  to 
its  surface.  The  intersection  of  the  wires  should  be 
brought  precisely  upon  the  object  to  which  the  telescope  is 
directed. 

The  zeros  of  the  verniers  and  limb  should  be  brought 
into  line  by  the  tangent  screw  of  the  leveling  head.  The 
angles  taken  are  then  read  off  upon  the  limb,  without  sub- 
tracting from  those  given  by  the  verniers  in  any  other 
position. 


TRANSIT  SURVEYING  223 

TO   ADJUST  THE  TRANSIT 

Every  instrument  should  leave  the  hands  of  the  maker 
in  complete  adjustment,  but  all  adjustments  are  liable  to 
derangement  by  accident  or  careless  use  so  it  is  necessary 
to  describe  particularly  those  which  are  most  likely  to 
need  attention. 

The  principal  adjustments  of  the  transit  are:  The 
Levels,  the  Line  of  Collimation,  the  Standards. 

To  adjust  the  levels.  —  Set  the  instrument  upon  its  tri- 
pod as  nearly  level  as  may  be,  and  having  undamped  the 
plates,  bring  the  two  levels  above,  and  on  a  line  with,  the 
two  pairs  of  leveling  screws.  Clasp  the  heads  of  two 
opposite  screws,  and,  turning  both  in  or  out,  as  may  be 
needed,  bring  the  bubble  of  the  level  directly  over  the 
screws  exactly  in  the  middle  of  the  opening.  Without 
moving  the  instrument,  proceed  in  the  same  manner  to 
bring  the  other  bubble  to  the  middle.  The  level  first  cor- 
rected may  now  be  thrown  a  little  out;  if  so,  bring  it  in 
again,  and  when  both  are  in  place  turn  the  instrument 
halfway  around.  If  the  bubbles  are  both  in  the  middle 
they  need  no  correction;  but  if  not,  turn  the  nuts  at  the 
end  of  the  levels  with  the  adjusting  pin,  until  the  bubbles 
are  moved  over  half  the  error,  bring  the  bubbles  again  into 
the  middle  by  the  leveling  screws,  and  repeat  the  oper- 
ation until  the  bubbles  will  remain  in  the  middle  during  a 
complete  revolution  of  the  instrument. 

To  adjust  the  line  of  collimation.  —  This  adjustment  is 
to  bring  the  cross-wires  into  such  a  position  that  the  in- 
strument, when  placed  at  the  middle  of  a  straight  line, 
will,  by  the  transit  of  the  telescope,  cut  the  extremities  of 
the  line.  Having  leveled  the  instrument,  determine  if  the 
vertical  wire  is  plumb,  by  focusing  on  a  defined  point  and 
observing  if  the  wire  remains  on  that  point  when  the  tele- 
scope is  elevated  or  depressed.  If  not,  loosen  the  cross- 
wire  screws  and  by  their  heads  turn  the  ring  until  correct, 
the  openings  in  the  telescope  tube  being  slightly  larger 
than  the  screws  so  that  when  the  latter  are  loosened  the 
ring  can  be  rotated  a  short  distance  in  either  direction. 
Direct  the  intersection  of  the  cross-wires  on  an  object  two 
or  three  hundred  feet  distant.  Set  the  clamps  and  transit 


224  PRACTICAL   SURVEYING 

to  an  object  about  the  same  distance  in  the  opposite  di- 
rection. Unclamp,  turn  the  plates  halfway  around,  and 
direct  again  to  the  first  object;  then  transit  to  the  second 
object.  (Note.  —  To  transit  is  to  revolve  the  telescope  on 
its  axis.  The  telescope  is  thus  reversed  to  get  a  sight  on 
an  object  on  the  line  in  rear  of  the  instrument.)  If  it 
strikes  the  same  place  the  adjustment  is  correct.  If  not,  the 
space  which  intervenes  between  the  points  bisected  in  the 
two  observations  will  be  double  the  deviation  from  a  true 
straight  line,  since  the  error  is  the  result  of  two  observations. 

In  Fig.  189  let  A  represent  the  center  of  the  instrument, 
and  BC  the  imaginary  straight  line,  upon  the  extremities 
of  which  the  line  of  collimation  is  to  be 
adjusted.     B  represents  the  object  first 
selected,   and  D   the  point  which  the 
wires  bisected  when  the  telescope  was 
F  reversed. 

When  the  instrument  is  turned  half 

around,  and  the  telescope  again  directed  to  B_,  and  once 
more  reversed,  the  wires  will  bisect  an  object  E,  situated 
as  far  to  one  side  of  the  true  line  as  the  point  D  is  on 
the  other  side.  The  space  DE  is  therefore  the  sum  of  two 
deviations  of  the  wires  from  a  true  straight  line,  and  the 
error  is  made  very  apparent. 

In  order  to  correct  it,  use  the  two  capstan-head  screws 
on  the  sides  of  the  telescope,  these  being  the  ones  which 
affect  the  position  of  the  vertical  wire.  It  must  be  kept 
in  mind  that  the  eyepiece  apparently  inverts  the  position 
of  the  wires,  and  therefore,  in  loosening  one  of  the  screws 
and  tightening  the  other  on  the  opposite  side,  the  operator 
must  proceed  as  if  to  increase  the  error  observed. 

The  wires  being  adjusted,  their  intersection  may  now 
be  brought  into  the  center  of  the  field  of  view  by  moving 
the  screws  holding  the  ring,  which  are  slackened  and 
tightened  in  pairs,  the  movement  being  now  direct,  until 
the  wires  are  seen  in  their  proper  position. 

The  position  of  the  line  of  collimation  depends  upon  that 
of  the  objective  solely,  so  that  the  eyepiece  may,  as  in  the 
case  just  described,  be  moved  in  any  direction,  or  even 
removed  and  a  new  one  substituted,  without  at  all  de- 
ranging the  adjustment  of  the  wires. 


TRANSIT  SURVEYING  225 

In  case  it  becomes  necessary  to  remove  the  cross-wire 
ring,  the  operator  should  proceed  as  follows:  Take  out  the 
eyepiece,  together  with  the  ring  by  which  it  is  centered, 
remove  two  opposite  cross-wire  screws,  sand  with  the  others 
turn  the  ring  until  one  of  the  screw  holes  is  brought  into 
view  from  the  open  end  of  the  telescope  tube.  In  this 
screw  hole  thrust  a  splinter  of  wood  or  a  wire  to  hold  the 
ring  when  the  remaining  screws  are  withdrawn.  The  ring 
can  then  be  removed.  It  may  be  replaced  by  returning 
it  to  its  position  in  the  tube,  and  after  either  pair  of  screws 
is  inserted  the  splinter  or  wire  is  removed,  and  the  ring  is 
turned  until  the  other  screws  can  be  replaced,  care  being 
taken  that  the  face  of  the  diaphragm  is  turned  toward  the 
eyepiece.  The  eyepiece  is  next  inserted,  and  its  centering- 
ring  brought  into  such  a  position  that  the  screws  in  it  can 
be  replaced,  and  the  ring  into  which  the  eyepiece  is  fixed 
is  then  screwed  to  the  end  of  the  telescope. 

To  adjust  the  standards.  —  In  order  that  the  point  of  in- 
tersection of  the  wires  may  trace  a  vertical  line  as  the 
telescope  is  elevated  or  depressed,  it  is  necessary  that  the 
standards  of  the  telescope  should  be  of  precisely  the  same 
height.  To  ascertain  this,  and  make  the  correction,  if 
needed,  proceed  as  follows: 

Having  the  line  of  collimation  properly  adjusted,  set  up 
the  instrument  in  a  position  where  points  of  observation, 
such  as  the  apex  and  base  of  a  lofty  spire,  can  be  selected, 
giving  a  long  range  in  a  vertical  direction. 

Level  the  instrument,  direct  the  telescope  to  the  top  of 
the  object,  and  clamp  to  the  spindle;  then  bring  the  tele- 
scope down  until  the  wires  bisect  some  well-defined  point 
at  the  base.  Turn  the  instrument  half  around,  direct  the 
telescope  to  the  lower  point,  clamp  to  the  spindle,  and 
raise  the  telescope  to  the  highest  point.  If  the  wires 
bisect  it,  the  vertical  adjustment  is  effected;  if  they  are 
thrown  to  either  side,  this  proves  that  the  standard  op- 
posite to  that  side  is  the  highest,  the  apparent  error  being 
double  that  actually  due  to  this  cause.  When  a  transit 
does  not  have  any  means  for  adjusting  the  height  of  the 
standard  it  should  be  sent  to  the  maker  for  adjustment. 

Cross-wires  are  usually  made  of  platinum.  Occasionally 
spider  webs  are  used  although  few  modern  instruments  for 


226  PRACTICAL  SURVEYING 

ordinary  work  are  so  equipped,  as  rough  usage  and  long 
periods  of  damp  weather  injure  spider  hairs.  When  it  is 
necessary  to  replace  a  broken  wire  or  hair,  hunt  for  the 
small  black  spider,  generally  found  in  trees  and  under- 
brush, for  all  other  spider  web  is  too  coarse.  Allow  the 
spider  to  run  out  on  a  pencil  or  small  stick.  When  the  end 
is  reached  it  attaches  a  line  and  drops,  thus  stretching  the 
line  taut.  The  diaphragm  having  been  removed  from  the 
telescope  and  the  broken  wires  removed,  grooves  made  by 
the  instrument  maker  will  be  found.  A  drop  of  shellac  is 
placed  in  each  groove,  and  the  stretched  web  fitted.  The 
spider  will  not  spin  when  cold  and  when  sulky  must  be 
tossed  and  rolled  gently  until  anxiety  to  escape  is  exhibited. 

CARE  OF  TRANSITS  AND   LEVELS 

Remarks  on  care  of  the  compass  apply  equally  to  the 
compass  attached  to  a  transit. 

All  instruments  should  be  protected  during  a  rain  by 
covering  with  a  bag.  For  a  transit  use  only  oiled  silk,  for 
oiled  linen  and  the  sulphur  in  rubber  will  blacken  the  silver 
on  the  horizontal  and  vertical  limbs. 

In  bright  sunshine,  on  a  hot  day,  it  is  a  sensible  idea  to 
shade  the  instrument,  to  avoid  errors  due  to  expansion. 
Expansion  sometimes  injures  the  finer  parts  of  an  instru- 
ment. The  use  of  a  bag  during  wet  weather  is  common 
but  very  few  surveyors  use  a  sunshade  on  hot  days,  a  habit 
all  should  acquire. 

Covering  an  instrument  during  a  fog  is  wise  and  cover- 
ing it  in  cold  weather  is  to  be  done  cautiously,  for  it  may 
" sweat"  with  the  frost.  A  soft  linen  cloth  should  be 
carried  and  the  instrument  wiped  often  to  remove  dust  and 
moisture.  When  dry  go  over  it  with  a  large,  soft  camels- 
hair  brush. 

Never  place  an  instrument  in  a  box  when  damp.  Be  sure 
it  is  dry  and  then  wipe  and  brush  it  before  locking  in  the 
box.  To  dry  an  instrument  place  it  in  a  dry,  warm  room. 

Never  leave  an  instrument  standing  in  an  open  space 
without  a  person  to  guard  it.  In  a  room  place  it  in  a  corner 
with  the  points  of  the  tripod  legs  set  in  floor  cracks.  An 
instrument  left  on  a  tripod  out  in  a  room  is  easily  knocked 


TRANSIT  SURVEYING  227 

over.  After  an  instrument  is  dry  and  has  been  cleaned, 
place  it  in  a  box. 

If,  after  exposure  of  an  instrument  in  extremely  hot  or 
cold  weather,  it  is  found  that  the  centers  do  not  revolve 
as  freely  as  usual,  .clean  them  as  soon  as  possible. 

In  cleaning  object  and  eyepiece  glasses,  use  a  soft  rag 
or  chamois  leather.  If  the  glasses  should  become  greasy 
or  very  dirty,  wash  them  with  alcohol.  As  the  fine  polish 
on  the  object  glass  will  be  destroyed  by  wiping  too  often, 
the  instrument  man  should  be  careful  in  this  respect. 
Should  telescope,  compass  or  vernier  glasses  become  moist, 
place  the  instrument  in  a  room  which  is  dry  and  moderately 
warm.  Should  it  be  impossible  to  follow  this  method,  the 
glasses  may  be  wiped  dry,  this  latter  process,  however, 
affording*  an  opportunity  for  dirt  and  dust  to  get  into  the 
instrument  while  the  glasses  are  removed.  The  inner 
surfaces  of  protected  glasses  need  seldom  be  wiped. 

Due  attention  should  be  given  the  screws  confining  tele- 
scope bearings.  They  should  be  tightened  sufficiently  to 
make  the  bearing  firm  and  still  permit  the  telescope  to  re- 
volve freely,  yet  be  kept  in  position  by  the  friction  thus 
obtained.  When  there  is  too  much  friction  in  the  telescope 
slide,  take  it  out  immediately  and  first  scrape  the  rough 
place  with  the  blade  of  a  pocket  knife,  having  its  edge  in- 
clined a  trifle,  then  use  the  back  of  blade  for  burnishing 
the  spot.  If  possible,  treat  inside  of  tube  in  a  similar 
manner,  or  at  least  wipe  it  out.  The  slide  should  then  be 
slightly  greased  with  watch  oil,  but  all  the  grease  must  be 
wiped  off  before  slide  is  replaced.  No  emery  or  emery 
paper  should  be  used  on  any  part  of  an  instrument.  When 
fretting  begins  it  should  be  sent  to  an  instrument  maker 
for  repairs.  If  this  cannot  be  done  promptly,  no  man 
should  be  afraid  to  take  an  instrument  apart  for  cleaning. 
It  is  of  course  foolish  to  take  an  instrument  apart  when  not 
necessary,  but  many  surveyors,  afraid  to  "examine  the 
insides,"  keep  on  using  a  dirty  instrument  until  it  gets 
into  such  bad  shape  that  an  instrument  maker  loses  money 
putting  it  into  shape  again. 

A  skilful  man  can  take  apart  and  put  together  an  in- 
strument without  damage;  while  an  unskilful  man  will 
certainly  do  harm  by  attempting  such  work. 


228  PRACTICAL  SURVEYING 

Vaseline  will  clean  the  surface  of  the  silvered  ring  on 
the  limbs  and  should  be  thoroughly  wiped  off.  Use  only 
good  watch  oil  for  lubricating  centers  and  cleaning  screw 
threads.  Use  as  little  as  possible  and  wipe  it  off  so  it  will 
not  gather  dust  and  grit.  Fretting  centers  should  be 
treated  in  the  same  manner  as  fretting  telescope  tubes. 

Overstraining  of  screws  should  be  guarded  against  as  it 
either  stretches  the  threads,  causing  them  to  wear  out  in  a 
short  time,  or  gradually  loosens  some  part  of  the  instru- 
ment. When  screws  are  too  tight  the  instrument  is  sen- 
sitive to  temperature  changes;  when  too  loose  it  is  unsteady 
and  reliable  work  cannot  be  done. 

Tripod  legs  must  be  firm.  If  the  screws  are  loose  the 
instrument  will  be  shaky  and  if  too  tight  warping  may 
cause  the  telescope  to  shift  off  the  line  of  sight.  The 
proper  degree  of  restraint  may  be  determined  by  raising 
the  legs  one  at  a  time  to  a  horizontal  position.  If  the 
screw  holds  a  leg  up,  it  is  too  tight.  If  the  leg  falls  rapidly, 
the  screw  is  too  loose.  The  downward  turning  should  be 
slow  and  uniform. 

TO  READ  ANGLES  WITH  A  TRANSIT 

The  instrument  is  set  at  B  (Fig.   190)  and  leveled  by 
means   of   the   leveling   screws.     All   clamp   screws   being 
loose,  bring  the  zeros  on   the  horizontal 
"A  limb  and  vernier  together  and  clamp  with 
the  upper  clamp  screw.      Then  use  the 
reading  glass  —  which  hangs  by  a  cord 
around  the  neck  —  and  with   the  upper 
FIG   loo       "  tangent   screw   bring    the    zeros    exactly 

together. 

Revolve  the  instrument  horizontally  on  its  vertical  axis, 
without  touching  the  upper  clamp  or  tangent  screw,  until 
the  vertical  cross- wire  bisects  the  stake  (or  tack)  at  A. 
Clamp  the  lower  clamp  screw  and  with  the  lower  tangent 
screw  bring  the  cross-hair  to  the  exact  point  desired. 

The  instrument  is  now  clamped  with  zeros  together  and 
cross-hairs  bisecting  the  object.  It  cannot  be  revolved 
on  the  vertical  axis.  Loosen  the  upper  clamp  screw  and 
direct  the  telescope  to  C.  Clamp  the  screw  and  by  means 


TRANSIT  SURVEYING  229 

of  the  upper  tangent  screw  bring  the  cross-hair  to  exactly 
bisect  the  point  at  C.  Now  with  the  reading  glass  the 
angle  may  be  read.  If  the  instrument  is  in  perfect  ad- 
justment, well  made,  with  no  errors  in  graduation,  and  it 
is  not  disturbed  during  the  operation,  the  angle  is  obtained 
correctly. 

To  check.  —  Unclamp  the  lower  clamp  screw  without 
touching  the  upper  screws.  Reverse  the  telescope.  Re- 
volve the  instrument  horizontally  and  bisect  A  with  the 
vertical  wire.  Clamp  the  lower  clamp  and  bring  the  wire 
to  an  exact  bisection  by  means  of  the  lower  tangent  screw, 
the  plate  meanwhile  remaining  set  at  the  angle  read. 
Unclamp  the  upper  screw  and  direct  the  telescope  to  C. 
Clamp,  and  with  the  upper  tangent  screw  bisect  C  with 
the  vertical  wire.  Now  read  the  angle,  which  should  be 
double  the  exact  angle.  The  operation  just  described  is 
called  "Double-centering."  The  mean  of  the  two  read- 
ings is  correct  because  every  error  in  one  direction  on  the 
first  reading  is  balanced  by  an  equal  error  in  the  opposite 
direction  on  the  second  reading.  This,  however,  does  not 
apply  to  errors  in  graduation.  When  running  lines  every 
angle  "on  lii\e"  should  be  "double-centered,"  but  only  one 
reading  is  necessary  for  side  shots  and  lines  where  the 
closest  possible  accuracy  is  not  required. 

Repeating  angles.  —  In  triangulation  work  the  method 
of  repetition  is  followed.  Read  the  angle  by  sighting  to 
A  and  then  to  C.  Keep  the  telescope  erect  and  repeat  the 
readings  from  A  to  C  until  a  total  of  six  are  taken.  The 
final  reading  divided  by  6  gives  an  approximate  value  for 
the  angle.  Both  verniers  are  read  at  the  start  and  after 
the  sixth  repetition.  The  total  reading  of  each  is  divided 
by  6.  If  there  is  a  difference  the  mean  of  the  final  values 
is  taken. 

Reverse  the  telescope  and  repeat  the  process,  but  this 
time  reading  from  right  to  left.  Read  both  verniers, 
divide  the  totals  by  6,  and  take  the  mean  value  of  the  six 
readings  of  each  vernier. 

We  now  have  a  mean  value  for  the  direct  readings 
and  one  for  the  reversed  readings.  The  mean  of  the 
means  should  be  the  true  angle,  plus  or  minus  a  probable 
error. 


230  PRACTICAL  SURVEYING 

Let  S  =  "the  sum  of  all  the  quantities  in  the  paren- 
thesis," 
v  =  variation  (or  deviation)  of  each  angle  from  the 

mean,  provided  each  angle  is  read, 
n  =  number  of  readings, 

r  =  probable  error, 
then 

r  =  d=  0.8453  ^  /    \       '  =.  for  single  observation, 
and 


r  =  ±  0.8453  —    ,  for  all  observations. 

n  Vn  —  i 

When  the  deviation  gives  a  larger  angle  than  the  mean 
it  is  positive  and  when  smaller  it  is  negative,  but  in  ob- 
taining the  sum  in  the  formulas  above  given  the  signs  are 
disregarded.  The  formulas  are  known  as  Peter's  Ap- 
proximations of  Bessel's  Formula,  which  is  based  on  the 
principle  of  least  squares  and  is  laborious  to  work. 

"The  probable  error  is  not  'the  most  probable  error,' 
nor,  'the  most  probable  value  of  the  actual  error.'  It  de- 
termines the  degree  of  confidence  we  may  have  in  using 
the  mean  as  the  best  representative  value  oT  a  series  of 
observations."  (Mellor.)  If  the  mean  value  of  the  angle 
is  17°  31'  42"  and  the  probable  error  is  5.33"  then  the 
odds  are  even  that  the  true  value  lies  somewhere  between 
17°  3i'  47-33"  and  17°  36'  36.67."  When  the  three  angles 
of  a  triangle  have  been  measured  and  the  sum  does  not 
equal  180°,  a  computation  of  the  most  probable  error  for 
each  angle  will  be  a  guide  to  balance  the  angles  to  obtain 
1 80°.  If  the  triangle  has  sides  more  than  a  mile  in  length 
it  may  be  the  error  in  closure  is  due  to  "spherical  excess" 
by  reason  of  the  curvature  of  the  earth,  the  triangle  being 
spherical.  Geodetic  methods  must  then  be  used. 

The  computation  for  probable  error  is  sometimes  used  in 
the  case  of  a  line  which  has  been  measured  several  times 
with  a  tape  forward  and  back  by  different  sets  of  chain- 
men.  A  small  probable  error  is  often  said  to  be  a  measure 
of  the  accuracy,  but  as  it  refers  only  to  the  proportion  in 
which  errors  of  different  magnitude  occur  it  cannot  be  a 
measure  of  accuracy. 


TRANSIT  SURVEYING  231 

Formulas  determining  the  probable  error  are  worthless 
for  a  small  number  of  observations.  We  may  select  ten 
as  the  least  number,  and  the  first  step  is  to  eliminate  all 
constant  errors.  If  a  tape  is  too  long  or  too  short  the  con- 
stant error  in  each  tape  length  is  the  difference  between  the 
length  of  the  tape  and  a  standard.  If  an  error  in  gradu- 
ation exists  in  some  part  of  the  circle  of  a  transit,  the  part 
in  which  the  error  exists  must  not  be  used.  In  chaining, 
three  sets  of  chainmen  should  measure  the  line  forward 
and  back  at  least  twice,  making  a  total  of  twelve  measure- 
ments. By  using  one  set  of  chainmen  the  personal  error 
would  not  be  eliminated,  and  as  it  is  a  constant  error  difficult 
to  find  and  evaluate,  other  men  must  be  used  to  give  this 
error  something  of  the  character  of  a  series  of  accidental 
errors.  The  careful  measuring  above  described  is  done 
only  on  the  highest  grade  work. 

Constant  errors  are  cumulative  and  accidental  errors 
are  compensating.  The  probable  error  is  practically  a 
mean  square  value  of  the  positive  and  negative  deviations 
from  the  arithmetical  mean  of  all  the  accidental  errors. 
The  probable  error  is  diminished  by  increasing  the  number 
of  observations. 

In  chaining,  the  accuracy  of  the  work  is  increased  by 
taking  plenty  of  time,  and  not  working  too  fast.  In  re- 
peating angles  with  a  transit  the  accuracy  of  the  work  is 
increased  when  speed  is  increased.  Clamping  and  un-, 
clamping  disturbs  an  instrument  and  may  throw  it  out  of 
level.  Standing  a  long  time  in  one  place  and  the  handling 
tend  to  make  one  of  the  metal  shod  tripod  legs  sink  into  the 
earth.  The  temperature  effects  may  be  equalized  in  rapid 
work,  for  one  side  may  be  unduly  heated  or  cooled  if  al- 
lowed to  remain  too  long  in  one  position. 

A  transit  should  always  be  tested  for  "index  error." 
This  may  be  due  to  faulty  graduation,  in  which  case  the 
instrument  should  be  returned  to  the  maker,  or  it  may  be 
caused  by  a  fall  bending  the  plate.  Even  straightening 
the  plate  may  not  restore  it,  and  the  surveyor  for  some 
good  reason  may  have  to  use  it  without  re-graduation. 

Set  three  stakes  with  tacks  in  the  top  so  the  small  acute 
angle  will  lie  between  10°  and  20°.  Start  from  o°  and  read 
the  angle  successively  around  the  circle  without  double 


232  PRACTICAL  SURVEYING 

centering.  This  must  be  done  rapidly.  The  angles  should 
be  equal.  When  a  place  is  found  where  the  reading 
differs  from  readings  preceding  or  following  it,  test  each 
degree  in  the  space  until  the  error  is  exactly  located. 
The  amount  does  not  matter  but  the  place  must  be  marked 
and  not  used  thereafter.  Sometimes  errors  of  very  small 
value  are  found  which  show  up  only  in  a  series  of  readings, 
and  are  distributed  around  the  circle.  Double  centering 
will  take  care  of  them. 

Unnecessary  stepping  around  an  instrument  must  be 
avoided.  A  good  instrument-man  never  "straddles"  a 
tripod  leg.  Transit  work  being  closer  than 
compass  work  every  instrument  station  must 
be  marked  with  a  tack.  The  plumb-bob 
hanging  from  the  center  of  the  vertical  axis 
must  be  centered  over  the  tack,  this  being 
accomplished  with  a  shifting  center.  The 
lower  end  of  the  axis  is  enlarged  in  a  spheri- 
cal shape  and  fits  in  a  spherical  socket  on  a 
small  plate,  thus  enabling  the  leveling  screws 
to  bring  the  axis  to  a  vertical  position  when 
the  tripod  head  may  not  be  level.  The  open- 
ing in  the  lower  plate,  which  is  screwed  to 
the  tripod  head,  is  larger  than  the  ball  but 
smaller  than  the  socket  plate.  By  loosening  the  screws 
the  center  may  be  shifted  to  carry  the  plumb-bob  to  any 
side.  When  finally  centered  the  screws  are  tightened  and 
the  plate  leveled. 

Plumb-bob  lines  must  be  lengthened  and  shortened  and 
a  number  of  devices  are  used  to  accomplish  this  without 
the  line  slipping  in  the  hook  at  the  upper  end.  Nearly  all 
catch  the  wind  and  are  objectionable  for  this  reason.  A 
knot  in  the  string  is  best,  but  when  loose  enough  to  allow 
the  cord  to  be  slipped  through  it  is  a  nuisance.  By  loop- 
ing the  cord  twice  around  the  hook  the  knot  may  be  loose 
and  the  cord  will  not  slip,  the  friction  of  the  loop  holding 
the  plumb-bob  in  place. 

In  sighting  lines,  steel  line  rods  are  used  and  it  takes 
considerable  experience  to  enable  a  helper  to  hold  the  rod 
vertical.  Sometimes  a  plumb  line  is  held  over  the  point 
but  no  man  has  so  steady  a  hand  that  confidence  can  be 


TRANSIT  SURVEYING  233 

had  in  such  a  sight.  Sometimes  a  line  rod  is  stuck  in  the 
ground  at  a  slant  so  it  is  above  the  tack.  To  this  the  plumb 
line  is  tied  and  moved  along  the  rod  until  the  point  of  the 
bob  is  centered  on  the  tack.  For  quicker  sights  the  helper 
places  one  knee  on  the  ground  and  rests  one  hand,  holding 
the  plumb  line,  on  the  other  knee.  The  plumb-bob  is  then 
centered  over  the  tack  and  the  white  line  is  viewed  with 
the  leg  as  a  background. 

A  target  of  white  celluloid  made  in  such  size  and  manner 
that  it  may  be  carried  in  the  pocket  and  attached  to  a 
plumb  line  when  needed  has  been  developed 
by  Kolesch  &  Company  of  New  York.  It 
is  circular  in  form  with  a  diamond-shaped 
cut-out,  which  offers  a  strong  contrast  to  the 
body  of  the  target.  It  can  be  moved  up  and 
down  the  plumb  line  at  will.  The  applica- 
tion of  this  device  will  be  recognized  by  all  sur- 
veyors who  have  taken  sights  against  a  back-  p 
ground  of  shubbery  or  buildings,  and  it  will 
enable  the  taking  of  longer  "shots."  It  has  the  advantage 
also  of  being  a  more  accurate  means  of  setting  a  true  point 
than  can  be  attained  with  a  flag  or  pole,  that  may  be 
slightly  out  of  plumb  when  the  sight  is  taken. 

A  piece  of  wood  about  the  width  of  a  lead  pencil,  sharp- 
ened on  the  lower  end  and  stuck  in  a  crack  behind  the 
tack  is  frequently  used  for  a  backsight. 

In  Fig.  190  let  the  transit  be  set  at  A  with  point  B  set. 
An  angle  is  to  be  laid  off  and  point  C  located.  Turn  off 
the  angle  and  set  a  stake  at  C.  The  helper  holds  the  rod 
on  the  stake  and  the  transitman  motions  right  or  left  un- 
til the  rod  is  in  line,  when  it  is  pressed  slightly  to  make  a 
mark.  The  telescope  is  reverse^,  the  lower  clamp  loosened, 
the  vertical  hair  set  on  B  and  the  lower  clamp  tightened. 
The  upper  clamp  is  loosened  and  the  telescope  directed  to 
C,  clamped,  and  the  upper  tangent  screw  used  to  obtain 
exactly  double  the  angle.  The  rod  is  now  directed  again 
to  line  and  another  mark  made.  The  true  point  lies  mid- 
way between  the  marks  and  a  tack  is  driven  to  fix  it. 

In  Fig.  193  is  shown  the  general  method  of  conducting 
a  transit  survey.  Assume  A  and  F  to  be  set.  The  transit 
is  set  up  at  A  with  horizontal  plate  and  vernier  clamped  at 


234  PRACTICAL  SURVEYING 

zero.     The  lower  clamp  is  loose.     Reverse  (transit)  the  tele- 
scope and  sight  F.     Clamp  the  lower  screw  and  with  the 
lower  tangent  screw  bisect  the  tack  at  F.     Transit  the  tele- 
scope  so  it  sights  forward  on  the  line  FA. 
Unclamp  the  plate,  turn  the  angle  to 
ifs.     the  right  and  set  point  B.      Move  to  B 
and  set  C  from  a  backsight  on  A.     Pro- 
ceed  around  the  field  in  this  manner,  fi- 
nally  closing  on  A ,  which  may  be  occupied 
a  second  time  to  check  the  traverse. 
IG'  I93'  In  the  example  given  all  the  angles  were 

to  the  right.  In  Fig.  194  the  angles  are  turned  to  the  right 
and  left,  but  the  work  is  the  same,  the  entries  in  the  field 
book  having  the  letter  R,  or  L,  as  the  case  may  be,  follow- 
ing the  amount  of  the  angle. 

The  needle  should  be  al- 
lowed to  swing  to  check  the 
angles.  This  is  a  rule  to  be 
followed  in  all  transit  work.  PIG  IQ4 

When   carrying   the   transit 

lift  the  needle  off  the  pivot  and  hold  against  the  glass  cover 
of  the  compass.  The  needle  reading  is  not  so  accurate  as 
angles  read  on  the  limb,  but  it  affords  a  check  so  a  mistake 
in  setting  down  R  for  L,  or  vice  versa,  is  quickly  discovered. 
Such  a  mistake  is  common. 

A  mistake  never  gets  past  the  person  making  it.  An 
error  is  a  mistake  discovered  by  others  and  reflects  upon 
the  ability  of  the  person  responsible.  All  the  field  and 
office  work  of  surveyors  must  be  checked  and  re-checked  so 
no  errors  can  be  found.  Checking  is  a  most  important 
duty  and  cannot  be  neglected. 

Angles  may  be  read  on  a  survey  by  setting  the  limb  and 
vernier  at  zero  on  each  station  and  double  centering  each 
time  on  important  work.  The  bearing  of  the  first  line  is 
set  down  in  the  field  book  and  all  calculations  of  bearings 
start  from  it  as  a  base.  The  following  rules  are  used  to  re- 
duce angles  to  bearings: 

Reduce  bearings  to  azimuth  angles,  as  follows: 

All  N  E  bearings  are  less  than  90°. 

S  E  bearings  lie  between  90°  and  180°  and  are  reduced  to 
azimuth  by  subtracting  from  180°. 


TRANSIT  SURVEYING  235 

S  W  bearings  lie  between  180°  and  270°  and  are  reduced 
to  azimuth  by  adding  to  180°. 

N  W  bearings  lie  between  270°  and  360°  and  are  re- 
duced to  azimuth  by  subtracting  from  360°. 

The  meridian  being  selected  as  an  azimuth  line  and  the 
meridian  bearing  of  the  starting  course  having  been  re- 
duced to  an  azimuth  bearing  by  the  above  method,  con- 
sider all  angles  to  the  right  as  positive  and  all  angles  to 
the  left  as  negative. 

Sum  less  than  90°.     Bearing  N  E. 

Sum  between  90°  and  1  80°.  Subtract  from  180°  and 
call  bearing  S  E. 

Sum  between  180°  and  270°.  Subtract  180°  and  call 
bearing  S  W. 

Sum  between  270°  and  360°.  Subtract  from  360°  and 
call  bearing  N  W. 

Example.  —  Starting  from  the  end  of  a  line  bearing 
S  1  8°  2  1  '  E  a  line  was  run  as  follows  : 

Feet 

0.  R  14°  13'  ......................................  280 

1.  R  74°  21'  ......................................  500 

2.  L  47°  11'  ..........................  :  ...........  320 

3.  Lio°o5'  ......................................  211 

4.  R  27°  09'  ......................................  419 

Find  the  bearings  of  the  lines,  referred  to  the  true 
meridian. 

179°  60'  Course  o.    179°  60' 

S     18°  21'  E  175°  52' 

161°  39'  =  Azimuth  of  reference  S      40°  08'  E 

0.  14°  13'  R     course. 

175°  52'  Course  I.    250°  13' 

1.  74°  21'  R  1  80°  oo/ 
249°  73'                                                          S      70°  13'  W 

2.  47°  n'  L 

202°  62'  Course  2.  203°  02' 


o       , 
219°  66' 


R  Course  3.    192°  57' 


S   12°  57' W 

4.  220°  06' 
I 80°  00 

S   40°  06' W 


Course  4.  220°  06' 
i 80°  oo 


236  PRACTICAL  SURVEYING 

From  a  study  of  the  foregoing  example,  the  following 
shorter  method  is  found: 

S  18°  2i'E 

14°  13'  R 
S  4°  08'  E 

74°  21'  R 
S  70°  13'  W 

47°  n'L 
S  23°  02'  W 

10°  05'  L 
Si2°  57' W 

27°  09'  R 
S  40°  06'  W 

The  student  may  reduce  angles  to  bearings  by  either 
method  as  the  example  given  illustrates  the  principle. 

The  reason  for  recording  the  needle  reading  each  time  is 
now  plain.  If  the  R  and  L  are.  always  recorded  properly 
the  needle  readings  will  check  the  computed  bearings 
within  a  few  minutes.  With  bearings  nearly  due  north 
and  south  or  east  and  west  the  check  may  not  be  valuable 
if  local  attraction  is  present,  but  it  should  never  be  omitted 
for  it  is  the  only  check  to  be  had  when  all  angles  start 
from  zero  and  are  read  right  and  left. 

Azimuth,  according  to  the  dictionary,  is  "the  angle  com- 
prised between  two  vertical  planes,  one  passing  through 
the  elevated  pole,  the  other  through  the  object."  Herschel 
counted  azimuth  from  o°  to  360°,  and  in  this  sense  it  is 
used  by  surveyors;  that  is,  it  indicates  all  angles  referred 
to  some  meridian  and  independent  of  bearings.  Bearings 
are  never  greater  than  90°  and  are  reckoned  to  the  east 
and  west  from  the  north  or  south  pole.  When  azimuth 
is  reckoned  from  the  north,  the  true  meridian  being  the 
azimuth  meridian,  it  is  easy  to  convert  azimuth  into  com- 
pass bearings,  as  in  the  example  just  given. 

There  is  good  reason  for  following  this  custom  as  com- 
pass bearings  have  been  used  many  centuries  because  of 
the  needle  pointing  north.  When  azimuth  is  reckoned 
clockwise  from  the  north  the  needle  readily  checks  all 
computed  bearings.  For  about  a  generation  a  number 
of  writers  of  surveying  textbooks  have  reckoned  azimuth 
from  the  south  clockwise  because  the  astronomical  pole  is 


TRANSIT  SURVEYING  237 

south.  This  may  be  scientifically  correct  for  a  very  few 
astrophysicists,  but  it  is  confusing  to  the  surveyor  to 
assume  one  pole  for  angle  work  and  another  pole  distant 
1 80°  for  work  involving  compass  bearings.  It  really 
makes  no  difference  to  the  educated  man  which  pole  he 
uses  but  the  surveyor  is  dealing  with  the  public  and  the 
public  always  "starts  from  the  north."  The  surveyor 
who  reckons  azimuth  from  the  south  must  use  two  standard 
meridians  and  some  day  will  make  mistakes.  The  writer 
one  summer  employed  a  young  graduate  who  reckoned 
azimuth  from  the  south  and  was  continually  mixing  his 
notes  until  he  abandoned  the  practice.  When  the  writer 
went  to  school,  in  the  8o's,  azimuth  was  any  angle  read 
clockwise  from  a  chosen  meridian  to  an  object,  so  that  an 
object  20°  L  would  have  an  azimuth  of  340°,  thus  prevent- 
ing errors  due  to  misplacing  the  letters  R  and  L;  no  hint 
was  given  that  azimuth  should  be  referred  to  the  south 
pole  or  the  north  pole. 

With  an  instrument  in  perfect  adjustment  and  handled 
by  a  competent  man  surveys  are  often  made  by  azimuth 
bearings  and  no  double  centering  is  done.  The  instrument 
is  set  up  at  the  starting  point,  the  variation  plate  indicat- 
ing the  correct  declination.  The  needle  is  let  down  and 
allowed  to  settle  so  it  points  to  the  true  north.  The  limb 
and  vernier  are  set  to  zero  and  the  instrument  clamped. 
The  needle  is  then  lifted  and  not  again  used. 

Unclamping  the  plates  the  telescope  is  directed  to  the 
next  station  and  the  angle  read  and  recorded  after  clamp- 
ing the  plates.  Carrying  the  instrument  to  the  next  sta- 
tion, the  telescope  is  reversed  and  sighted  to  the  station 
just  left;  and  the  instrument  clamped  below.  The  tele- 
scope is  transitted  and  then  read  forward  on  the  line, 
the  limb  being  still  set  at  the  azimuth.  Unclamping  the 
plate  the  telescope  is  directed  to  the  next  station  and  the 
azimuth  read  and  recorded.  These  operations  are  re- 
peated at  each  station  and  when  the  first  station  is  again 
occupied  and  a  sight  taken  to  the  second  station  the  first 
azimuth  should  be  obtained.  Actually  there  may  be. a 
difference  of  one  or  two  minutes,  which  may  be  distributed. 
On  a  closed  traverse  errors  in  reading  azimuth  tend  to 
compensate,  for  the  right  and  left  readings  balance;  but 


238  PRACTICAL  SURVEYING 

on  other  surveys  errors  are  apt  to  be  cumulative.     Azimuth 
notes  for  the  last  example  will  read  as  follows: 

Reference  line  161°  39'  Needle  S  18°  21'  E 

0.  175°  52'      280  ft. 

1.  250°  13'      500  ft. 

2.  203°  02'         320  ft. 

3.  192°  57'      211  ft. 

4.  220°  06'         419  ft. 

All  angles  are  azimuth  angles  on  stadia  surveys.  Not 
all  surveyors  double  center  their  angles,  a  surprisingly  large 
number  using  the  azimuth  method  exclusively. 

When  surveying  cheap  land  or  doing  work  requiring  no 
greater  accuracy  than  may  be  obtained  with  a  compass, 
the  transit  may  be  used  as  a  compass  and  all  readings 
taken  with  the  needle. 

A  very  accurate  method  for  setting  off 
an  angle  is  illustrated  in  Fig.  195. 

On  a  given  line  are  set  two  hubs  A  and 
B.  A  line  is  to  run  from  B  through  a  point 
c  D  and  this  line  must  be  laid  off  correctly. 
Setting  on  B  the  telescope  is  directed  to  A , 
FIG.  195.  "  plates  clamped  to  zero  and  an  angle  ABC 
read,  one  minute  smaller  than  the  angle 
ABD,  and  stake  C  set.  A  tack  is  placed  in  C  and  the 
angle  ABC  is  read  by  repetition  to  obtain  the  exact  value. 
The  length  BC  is  measured  with  the  greatest  possible 
accuracy.  The  exact  value  of  angle  ABC  is  subtracted 
from  the  angle  ABD  and  the  tangent  of  the  difference 
multiplied  by  the  length  BC  gives  the  small  distance  CD, 
which  may  be  measured  off  and  a  tack  set.  Or  the  angle 
ABD  may  be  read  and  a  broad  stake  driven  at  D  with 
a  tack  in  the  top.  The  angle  is  then  read  by  the  repeti- 
tive method  and  if  a  difference  is  found  from  the  true 
angle  a  correction  may  be  measured  and  the  tack  set  in  the 
proper  place. 

A  transit  may  be  set  on  a  line  between  two  points,  by 
setting  as  nearly  as  possible  on  line  and  leveling  the  in- 
strument. Then  sight  on  one  point,  clamp  the  plates  and 
reverse  the  telescope  for  a  sight  on  the  other  point.  If  it 
does  not  strike  it,  shift  the  transit  and  try  again.  A  skil- 
ful instrument-man  can  get  on  a  line  in  three  trials.  After 


TRANSIT  SURVEYING  239 

getting  on  line  reverse  the  telescope  for  a  sight  each  way, 
to  eliminate  errors  in  adjustment  and  disturbance  due  to 
handling. 

Fig.   196  illustrates  the  operation,  A   and  B  being  the 
points  and  C  the   transit  sta- 
tion, the  small  circles  near   C    o ^ § 

showing  trial  set-up  points.  F      i  6 

To  prolong  a  line,   the   ap- 
proved method  is  to  set  on  a  stake  and  take  a  backsight 
with  telescope  reversed  on  a  stake  on  line  in  the  rear. 

Transit  the  telescope  and  set  a  stake  ahead.  Then  double 
center  on  the  stake  and  set  a  tack  between  the  two  marks. 
Move  to  the  new  stake  and  repeat  for  the  next  one. 

To  set  a  line  of  stakes  between  two  stakes  set  on  one, 
sight  to  the  other  and  clamp  the  plates.  The  line  of  sight 
being  fixed  the  telescope  is  plunged  *  on  the  horizontal  axis 
and  the  cross-hairs  set  on  the  line  rod  used  to  locate  the 
positions  of  the  intermediate  stakes  and  tacks  are  driven  in 
them. 

A  good  compass  is  graduated  to  J  degree  and  by  using 
care  one-half  this  angle  may  be  estimated.  If  this  is  done 
the  maximum  error  in  angle  will  be  4  minutes,  of  which  the 
natural  tangent  is  0.00116.  Few  surveyors  read  a  bearing 
closer  than  J  degree  (15')  in  which  case  the  maximum 
error  will  be  8  minutes,  of  which  the  natural  tangent  is 
0.00233. 

Ordinary  transits  read  by  means  of  verniers  to  single 
minutes  so  the  maximum  error  in  angle  cannot  exceed  31 
seconds,  of  which  the  natural  tangent  is  practically  0.000146. 
It  is  not  uncommon  for  transits  to  be  graduated  to  read 
angles  as  small  as  30,  20  or  10  seconds,  while  if  angles  are 
read  by  the  repetitive  method  the  exact  size  of  an  angle 
may  be  ascertained  to  the  nearest  5  seconds  with  a  vernier 
reading  to  one  minute,  or  to  the  nearest  second  with  a 
vernier  reading  to  20  seconds. 

With  a  transit  a  skilful  instrument-man  can  do  all  the 
angular  work  with  as  high  a  degree  of  accuracy  as  skilled 
chainmen  can  do  the  measuring  with  first-class  tapes. 

Methods  for  transit  surveying  are  the  same  as  methods 

*  To  plunge  a  telescope  is  to  turn  it  on  its  axis  to  set  a  line  ahead. 
The  telescope  is  clamped  and  is  not  inverted  or  reversed  for  a  rear  sight. 


240  PRACTICAL  SURVEYING 

for  compass  work  plus  the  increased  accuracy.  The 
angles  right  and  left,  or  the  azimuth  bearings,  are  reduced 
to  compass  bearings  and  the  computations  for  area  are 
the  same  as  those  described  in  the  chapter  dealing  with 
the  compass.  Computations  for  traverse  work  and  for 
supplying  omissions  are  described  in  the  chapter  on 
trigonometry. 

Surveys  are  made  for  one  of  three  purposes,  sometimes 
for  all: 

1.  To  describe  boundaries. 

2.  To  obtain  areas. 

3.  To  make  a  map. 

Boundary  surveys  are  usually  made  by  running  out  the 
boundary  lines,  offsetting  a  few  feet  when  the  boundary  is 
obstructed  with  hedges,  fences,  etc. 

Area  surveys  not  calling  for  a  description  may  be  made 
by  methods  already  described  for  chain  surveys.  The 
oldest  book  on  surveying  of  which  we  have  knowledge, 
Hero's  Dioptra,  shows  that  the  custom  fully  twenty  centu- 
ries ago  was  to  lay  off  in  the  field  a  rectangular  figure  and 
erect  on  the  sides  of  it,  at  regular  intervals,  perpendiculars 
to  strike  the  boundary  lines.  The  areas  of  all  the  divisions 
were  added. 

When  the  diopter  was  improved  so  angles  other  than 
right  angles  could  be  turned  off,  surveyors  turned  angles 
from  points  on  the  boundary  of  the  interior  rectangle  to 
corners  of  the  field  and  measured  to  the  corners.  The 
proceeding  was  duplicated  in  making  the  map  and  the 
areas  of  all  the  interior  figures  were  computed  by  men- 
suration and  added.  With  the  introduction  of  plane 
trigonometry  and  the  use  of  the  compass  the  "radiation" 
method  became  standard  and  areas  were  computed  by 
trigonometric  methods.  Sometimes  the  surveyor  merely 
laid  off  a  carefully  measured  base  line  and  from  the  ends 
and  some  intermediate  points  took  bearings  to  corners  and 
measured  the  distances.  This  divided  the  field  into  tri- 
angles, the  areas  of  which  were  computed  trigonometri- 
cally.  When  the  "double  meridian "  method  for  computing 
areas  was  developed  in  Ireland  by  Thomas  Burgh,  the 
custom  came  in  of  running  out  the  exterior  lines.  The 
reason  this  was  considered  advisable  was  to  save  labor. 


TRANSIT  SURVEYING  241 

Prior  to  Burgh's  discovery  lines  were  run  around  the 
boundary  to  obtain  data  for  writing  a  legal  description, 
but  surveys  to  compute  areas  were  still  made  by  mensu- 
ration and  trigonometric  methods.  When  the  double 
meridian  distance  method  for  computing  areas  made  it 
possible  to  use  the  boundary  survey  data,  only  the  one 
survey  was  necessary. 

With  the  passing  years  the  "radiation"  method  of  sur- 
veying land  received  less  attention  in  textbooks,  although 
all  practical  surveyors  used  the  method  where  it  proved  to 
be  time  saving.  The  writer  had  to  make  several  surveys 
by  this  method  when  at  school  and  used  it  frequently  in 
practical  work. 

The  three  approved  methods  of  surveying  a  piece  of 
land  are: 

1.  The  traverse  method. 

2.  The  radiation  method. 

3.  The  intersection  method. 

In  the  1912  Proceedings  of  the  Illinois  Society  of  En- 
gineers and  Surveyors,  a  paper  by  Mr.  G.  W.  Pickets,  then 
an  instructor  in  Civil  Engineering  at  the  University  of 
Illinois,  compared  the  three  methods. 

The  results  were  as  follows: 

Speed  in  chaining.  —  Intersection  method,  first;  radi- 
ation method,  second;  traverse  method,  third.  In  most 
instances  the  radiation  method  called  for  20  per  cent  less 
chaining  than  the  traverse  method. 

Speed  in  instrument  work.  —  Intersection  method,  first; 
radiation  method,  second;  traverse  method,  third. 

Speed  in  computing  results. — Traverse  method,  first; 
radiation  method,  second;  intersection  method,  third. 

The  above  comparisons  prove  the  traverse  method  to  be 
the  more  expensive,  for  slow  work  in  the  field  means  the 
employment  of  a  party  of  men  for  a  longer  time.  A  little 
extra  time  in  the  office  usually  affects  one  man  only. 
Accurate  work  involves  careful  checking  and  half  the  work 
of  engineers  and  surveyors  seems  to  consist  in  checking 
lengths,  angles  and  computations. 

Comparison  of  accuracy.  —  Assuming  that  no  blunders 
are  made,  any  one  of  the  three  methods  is  accurate  enough 
for  all  practical  work.  With  respect  to  actual  accuracy 


242  PRACTICAL  SURVEYING 

the  intersection  method  is  first;   radiation  method,  second; 
traverse  method,  third. 

The  traverse  method  consists  in  running  out  the  bound- 
aries; or  running  on  offset  lines  as  closely  as  possible  to 
the  boundaries.  To  guard  against  error  in  reading  the 
needle,  or  setting  down  the  wrong  angle,  it  is  a  common 
practice  to  read  an  angle  or  take  a  bearing  to  some  one 
object  from  each  station. 

In  Fig.  197  bearings  were  taken  to  one  corner  of  the 
chimney  of  the  house  in  the  field.  When  making  the  map 

these  bearings  are  all  drawn  and 
should  intersect  at  a  common 
point.  This  method  has  always 
been  used  by  careful  surveyors 
on  compass  surveys.  Modern 
transits  are  equipped  with  stadia 
wires,  or  with  gradienters,  and 
distances  should  be  checked  with 
one  or  the  other.  In  this  one 
instance  the  gradienter,  if  new 
with  sharp  clean  threads,  has  an 
advantage  over  the  stadia.  The  gradienter  can  be  used 
if  one  foot  of  rod  can  be  seen,  while  the  stadia  requires  a 
graduated  rod.  If  a  plain  rod,  however,  is  used  the  sur- 
veyor can  direct  the  horizontal  cross-hair  to  one  mark  and 
read  the  vertical  angle.  Then  drop  the  wire  two,  three,  or 
more  feet  on  the  rod  and  read  a  second  angle.  One-half 
the  sum  of  the  two  angles  gives  the  vertical  angle  to  the 
mid-point,  the  rod  being  held  vertically. 

Let   A  =  half  the  sum  of  the  two  angles, 

D  =  distance  on  rod  intercepted  between  the  wire 

on  the  two  sights  in  feet, 
L  =  length  of  line  in  feet, 
then      L  =  D  X  Cos2 A  +  I. 

The  formula  is  only  closely  approximate,  but  errors  in 
measuring  within  reasonable  limits  are  readily  caught  by 
such  a  check. 

In  the  radiation  method  the  transit  is  set  up  at  one  or 
more  points,  from  which  angles  and  lines  are  measured  to 
the  corners  of  the  field. 


TRANSIT  SURVEYING  243 

Fig.  198  illustrates  a  field  of  less  than  160  acres  in  fairly 
compact  shape.  Here  all  the  corners  could  be  seen  from 
one  set-up.  To  survey  the  field  by  the  traverse  method 
would  call  for  the  measurement  of  the  five  sides  and  for  the 
occupation  of  each  corner  with  the  instrument.  The  sum 
of  the  lengths  of  the  lines  OA  +  OB  +  OC  +  OD  +  OE 
is  usually  less  than  the  perimeter  of  the  field,  but  all 
lengths  should  be  checked.  Surveyors  skilled  in  stadia 
work  can  dispense  with  the  chain  or  tape. 


FIG.  198.  FIG.  199. 

Fig.  199  illustrates  a  larger  field,  the  corners  of  which 
cannot  be  seen  from  one  point.  The  instrument  is  set  at 
O  and  P,  the  angles  at  each  point  being  turned  from  a 
backsight  on  the  other.  All  the  radiating  lines  are  meas- 
ured except  PB  and  PE. 

Fig.  200  illustrates  a  long  narrow  field  and  neither  point 
0  nor  P  is  visible  from  the  other.  Flags  are  set  on  the 
boundaries  at  E  and  F  and  angles  read  to  them,  the  lengths 
OE,  OF,  PE  and  PF  being  measured. 


V    ^: 

/'N 


p 


FIG.  200.  FIG.  201. 


Fig.  20  1  illustrates  the  method  to  use  with  fields  of  con- 
siderable size,  say  exceeding  320  acres,  in  which  the  method 
shown  in  Fig.  200  may  not  be  applicable.  A  small  traverse 
is  run  within  the  field  and  angles  and  distances  measured 
to  the  corners  of  the  field. 

The  radiation  method  divides  a  field  into  triangles,  two 
sides  and  the  included  angle  of  each  being  measured. 


244  PRACTICAL   SURVEYING 

The  area  of  each  triangle  =  \  be  Sin  A  and  the  sum  of  the 
areas  of  the  triangles  equals  the  area  of  the  field,  Fig.  201 
being  an  exception.  If  a  description  of  the  field  is  wanted 
each  triangle  must  be  solved  for  the  third  side  and  the  base. 
"The  intersection  method  consists  in  establishing  and 
carefully  measuring  a  base  line,  from  each  end  of  which 
all  the  corners  of  the  field  are  visible,  and  in  measuring  all 
the  angles  around  each  end  formed  by  lines  radiating  to 
the  corners.  If  the  field  is  a  small  one  and  topography 
permits,  one  base  line  is  sufficient,  but  in  the  case  of  a 
large  field,  two  or  more  base  lines  will  be  necessary.  The 
base  line  is  taken  inside  the  field  when  it  is  possible,  as  the 
lengths  of  the  sights  is  thereby  reduced  to  a  minimum;  but 
it  is  not  essential  that  it  be  so  taken,  and  it  may  be  all  out- 
side of  the  field  or  partly  inside  and  partly  outside.  The 
only  imperative  condition  —  using  one  base  line  —  is  that 
all  corners  must  be  visible  from  each  end  of  the  base  line." 
"In  locating  the  base  line  care  must  be  taken  to  avoid 
small  angles,  as  the  sines  of  small  angles  change  rapidly. 
(Angles  under  15°  and  over  165°  must  be  obtained  to  the 
nearest  10  sees.,  which  can  be  done  by  trebling  the  angle 
on  the  limb  of  the  transit.)  In  the  case  of  a  rectangular 
field  the  best  location  for  the  base  line  is  parallel  to  the 
short  axis.  This  arrangement  should  be  followed  as  closely 
as  the  topography  will  permit.  The  more  irregular  the 
field,  the  harder  it  is  to  avoid  small  angles,  and  more  time 
and  thought  are  required  in  the  selection  of  the  base  line 
than  in  the  case  of  the  rectangular  field.  The  length  of 
the  base  line  depends  upon  local  conditions.  In  general, 
the  longer  it  is,  the  more  accurate  the  results;  but  it  should 
never  be  less  than  one-third  the  length 
of  the  average  sight.  This  is  based  on 
the  assumption  that  the  instrument  used 
reads  to  30  sees.,  and  that  each  angle  is 
doubled  on  the  limb." 

Fig.  202  illustrates  a  field  surveyed  by 
FIG   202  ^ne  intersection   method,  AB  being  the 

base  line,  the  only  line  measured. 
"The  traverse  method  is  applicable  to  any  field  regard- 
less of   the  crops  or  topographical  conditions.     There  is 
always  a  strip  of  land  along  the  fence  line  that  is  not  culti- 


TRANSIT  SURVEYING  245 

vated,  and  an  open  sight  may  be  had  along  this  strip,  so 
the  distance  between  corners  may  be  chained  without 
interfering  with  the  growing  crops.  The  radiation  method 
may  be  used  equally  as  well,  provided  that  the  growing 
crops  do  not  interfere  with  the  chaining  or  obscure  the 
lines  of  sight.  Farms  are  generally  surveyed  upon  chang- 
ing hands,  and  the  principal  thing  to  be  ascertained  is  the 
acreage.  Possession  is  taken  in  the  spring  before  the  crops 
are  put  in,  and  hence  the  surveyor  is  not  limited  in  his 
choice  of  methods.  The  intersection  method  can  be  used 
only  in  level,  open  country,  as  all  the  corners  of  the  field 
can  seldom  be  seen  from  both  ends  of  a  base  line  in  hilly 
wooded  country.  Hence  with  respect  to  applicability,  the 
traverse  method  has  a  slight  advantage  over  the  radiation 
method,  and  both  of  these  are  far  ahead  of  the  inter- 
section method." 

"Which  method  of  surveying  should  be  used  in  any  given 
case  depends  entirely  upon  the  conditions  peculiar  to  that 
particular  case.  Each  of  the  methods  is  best  in  some  in- 
stances. The  use  of  the  intersection  method  is  limited,  it 
is  true,  by  topographical  features,  but  it  is  the  most  ac- 
curate of  all,  and  should  be  used  when  the  nature  of  the 
country  will  permit.  The  radiation  method  should  be 
given  equal  standing  with  the  traverse  method,  for  its 
saving  in  time  more  than  compensates  for  its  slightly 
limited  use.  Frequently  the  sides  of  a  field  have  to  be 
measured  in  order  to  re-locate  the  corners,  and  in  such 
cases  the  traverse  method  is  the  quickest  and  should  be 
used.  As  any  one  of  the  methods  is  accurate  enough  for 
all  practical  purposes,  the  one  that  is  most  applicable  to 
the  case  in  hand  should  be  used." 

Figs.  198  to  202  inclusive,  and  the  quoted  paragraphs  are 
from  the  paper  by  Mr.  Pickels. 

ANGULAR  LEVELING 

The  long  level  under  the  telescope  of  a  transit  may  be 
used  for  leveling,  as  already  described,  or  vertical  angles 
may  be  read  and  the  rise  or  fall  computed  by  the  formula 

H  =  ±  I  tan  A, 
in  which  H  =  height  or  difference  in  elevation  in  feet, 

/  =  length  of  sight  in  feet  (horizontal), 
A  =  angle  of  elevation  or  depression. 


246  PRACTICAL  SURVEYING 

The  intersection  of  the  cross-hairs  should  be  directed  to 
a  point  at  their  height  above  the  ground.  A  rod  is  generally 
used  to  sight  on.  The  horizontal  distance  must  be  carefully 
measured.  The  vertical  angle  should  be  read  at  both 
stations,  being  an  angle  of  elevation  (  +  )  at  one  and  an 
angle  of  depression  (  —  )  at  the  other.  If  there  is  a  small 
difference  use  the  mean  value.  A  large  difference  calls  for 
a  check  and  may  be  due  to  lack  of  adjustment.  The  long 
bubble  should  be  adjusted  so  it  will  remain  centered  as  the 
instrument  revolves  on  its  vertical  axis.  The  cross-hairs 
should  then  be  adjusted  as  described  for  the  dumpy  level, 
so  the  line  of  sight  will  be  parallel  to  a  horizontal  line  as 
indicated  by  the  bubble.  The  zero  of  the  vernier  should 
be  adjusted  to  coincide  with  the  zero  of  the  vertical  arc. 

Leveling  in  mining  surveys  is  generally  angular  work 
underground.  On  steep  ground  and  in  rolling  country 
leveling  by  vertical  angles  is  rapid  and  when  care  is  used 
is  very  accurate.  It  is  not  well  adapted  for  work  requir- 
ing elevations  at  intervals  of  about  100  ft.  or  less. 

STADIA  WIRES 

When  equipped  with  stadia  wires  so  distances  may  be 
read  on  a  rod,  the  transit  becomes,  as  it  has  been  adver- 
tised, "A  universal  surveying  instrument." 

Stadia  wires  are  two  horizontal  wires,  one  above  and 
one  below  the  horizontal  wire  found  in  all  transit  tele- 
scopes. These  two  wires  are  fixed  in  a  ring  or  diaphragm 
in  such  a  way  that  the  interval  usually  intercepts  a  space 
of  one  foot  on  a  rod  held  at  a  point  distant  d  ft.  from 
the  center  of  the  transit,  plus  a  small  constant  C. 

In  Fig.  203  the  relations  are  shown: 

c  =  distance  from  telescope  axis  to  center  of  objective, 
/  =  focal  length,  the  distance  from  the  stadia  wires  to 

the  objective, 
D  =  d  +  C    and     C  = 


The  rays  cross  each  other  so  that  the  vertex  of  the  visual 
angle  is  not  at  the  center  of  the  instrument,  but  at  a  dis- 
tance in  front  of  the  objective  equal  to  its  focal  length, 
measured  when  the  telescope  is  focused  on  a  distant  object. 


TRANSIT  SURVEYING 


247 


The  relation  between  the  size  and  distance  of  an  object 
and  the  size  of  its  image  in  a  telescope  is  given  by  the 
formula 

&_f  ,  _fR 

R~  d  =  R1' 

in  which  R  =  space  on  rod  intercepted  between  the  two 

wires  and  called  the  "rod  reading," 
R1  =  space  between  the  wires, 
d  =  distance  from  principal  focal  point  to  rod. 


.Stadia  wires 

—    ••••]  ^objective 


FIG.  203. 


When  the  wires  are  set  with  a  space  =  of  the  focal 

100 

length  between  them  we  have  the  relation  —=•  = ,  so 

the  wires  will  read  one  foot  on  a  rod  held  at  a  distance  of 
100  ft.,  one  yard  at  a  distance  of  100  yds.,  one  meter  at 
a  distance  of  100  m.,  etc.  Substituting  this  value  in  the 
above  expression  d  =  100  R,  to  which  must  be  added  the 
constant  C. 

This  constant  never  varies  for  any  given  instrument  and 
is  independent  of  the  distance.  Focus  the  telescope  on  an 
object  several  miles  away  and  measure  from  the  axis  to 
the  objective  for  c\  and  from  the  screws  holding  the  stadia 
diaphragm  to  the  objective  for/.  The  constant  C  =  c  +/ 
is  noted  by  all  instrument  makers  on  a  card  or  label  affixed 
to  the  box  of  every  instrument  provided  with  stadia  wires. 

When  using  the  stadia  the  rod  is  held  vertically.  The 
middle  wire  is  directed  to  a  point  on  the  rod  at  the  same 
height  above  the  ground  as  the  axis  of  the  telescope.  The 
difference  between  readings  of  the  upper  and  lower  wires 
gives  the  rod  reading,  if  a  level  rod  is  used,  or  a  rod  having 


248  PRACTICAL  SURVEYING 

numbers  on  the  face.     Regular  stadia  rods  are  seldom  num- 
bered, the  intercepted  space  being  read  directly. 

The  rod  is  held  vertically  and  as  the  line  of  sight  is 
seldom  horizontal  this  introduces  relations  between  the 
resulting  angles  which  are  finally  resolved  into  the  follow- 
ing expressions: 

D  =  Rcos2A  +  CcosA, 

„.       D  sin  2  A    .         . 

H  =  R  -        — h  C  sin  A , 

in  which  D  =  horizontal  distance  from  center  of  transit  to 

rod, 

R  =  rod  reading, 
A  =  vertical  angle, 
C  =  c+f, 
H  =  height  or  difference  in  elevation. 

For  ordinary  work  add  C  instead  of  C  cos  A ,  to  obtain 
horizontal  distance;  and  add  0.015  ^4  instead  of  C  sin  A, 
to  obtain  difference  in  elevation.  The  tables  here  given 
save  a  great  deal  of  time  in  reducing  inclined  stadia  read- 
ings to  horizontal  and  vertical  equivalents. 

Example. — An  angle  +18°  12'  was  read  to  a  rod  with 
a  rod  reading  of  5.72  ft.  Find  horizontal  distance  and 
difference  in  elevation. 

Turning  to  column  headed  18°  on  the  line  opposite  12' 
find:  Hor.  Dist.  =  90.24,  Diff.  Elev.  =  29.67.  At  the 
bottom  find  the  values  for  C  =  —  0.95  and  0.32,  for  we 
here  assume  C  =  i.oo  ft. 

Hor.  Dist.  =  (5.72  X  90-24)  +  0.95  =  517.12  ft. 

Diff.  Elev.  =  (5.72  X  29.67)  +  0.32  =  170.03  ft. 

A  number  of  excellent  labor-saving  slide  rules  and  dia- 
grams are  sold  by  instrument  dealers  for  reducing  stadia 
rod  readings. 

When  running  lines  with  the  stadia,  readings  should  be 
taken  forward  and  back  for  a  check  both  for  angle  and  rod 
reading.  Angles  should  be  kept  below  20°  when  possible 
and  readings  distorted  by  heat  are  unreliable.  When  con- 
ditions are  favorable  and  the  surveyor  is  careful  the  limit 

of  error  can  be  kept  within  —  — ,  a  degree  of  accuracy  supe 

1500 

rior  to  ordinary  farm  survey  chaining. 


TRANSIT  SURVEYING  249 

The  ratio  I  :  100  is  convenient  for  the  reason  that  ordi- 
nary level  rods  may  be  used  and  also  the  decimally  divided 
rods  sold  by  all  dealers.  If  some  other  ratio  is  used  the 
rods  must  be  specially  divided  or  all  reduced  readings  be 
multiplied  by  a  constant. 

Instrument  dealers  advise  the  use  of  fixed  wires  and  so 
do  men  in  the  government  service  who  are  constantly 
employed  on  stadia  work.  The  author  however  worked 
for  a  number  of  years  in  regions  where  instrument  repair 
shops  were  many  days  distant  and  not  all  instrument  re- 
pairers were  first  class.  Sudden  changes  in  temperature 
and  long-heated  terms  disturbed  the  wire  interval  so  many 
times  that  finally  adjustable  wires  were  placed  in  his  tran- 
sits and  all  trouble  ceased.  He  believes  that  "all  surveyors 
similarly  circumstanced  should  use  adjustable  stadia  wires 
and  this  after  an  extended  experience  in  stadia  work. 

To  adjust  the  wires,  select  a  piece  of  ground  practically 
level  and  lay  off  on  it  with  a  steel  tape  an  accurately  meas- 
ured line  several  hundred  feet  long  plus  C.  Set  the  transit 
at  one  end  and  have  the  rod  held  vertically  at  the  other  end. 
Level  the  telescope  and  sight  on  the  rod.  Deduct  C  from 
the  exact  distance  and  with  the  adjusting  screws  set  the 
upper  and  lower  wires  to  intercept  this  interval  on  the 
rod,  taking  care  to  keep  them  equidistant  from  the  middle 
wire.  Assume  that  the  transit  is  701  ft.  from  the  rod  and 
C  =  i.o  ft.;  the  rod  reading  will  be  7  ft.  To  catch  changes 
in  the  wire  interval  the  writer  checked  with  a  steel  tape  the 
first  and  last  rod  reading  in  the  forenoon  and  the  first  and 
last  rod  reading  in  the  afternoon  of  each  day.  Frequently 
the  wires  held  the  proper  interval  for  6  months  and  once 
no  change  was  observed  for  a  year.  When  the  interval 
did  change  it  was  sudden  and  the  ratio  became  as  much  as 
i  :  104,  generally  however  being  I  :  102.5  or  l  '•  IO3-3»  or 
a  similarly  annoying  ratio.  Instrument  repairers  generally 
blamed  the  cement  used  to  hold  the  wires  in  place.  Some- 
times temperature  effects  on  the  diaphragm  or  too  tight 
screwing  up  of  adjusting  screws  may  cause  these  alter- 
ations in  the  wire  interval,  which  have  been  noticed  by 
other  writers.  Alterations  due  to  such  .causes  disappear 
within  a  short  time  and  the  interval  is  apparently  correct. 
By  checking  rod  readings  occasionally  with  a  steel  tape 


250  PRACTICAL  SURVEYING 

and  correcting  readings  by  applying  a  constant  thus  found 
the  stadia  becomes  one  of  the  most  useful  of  the  tools  of 
the  surveyor.  (See  pp.  1041-2,  Vol.  LXXVII,  Trans.  Am. 
Soc.  C.E.) 

Stadia  surveys  of  farms  are  made  with  a  degree  of  ac- 
curacy suitable  for  the  work.  Either  the  radiation  or  the 
traverse  method  may  be  used,  or  a  combination  of  these 
methods.  Because  the  vertical  angle  must  be  read  to 
reduce  the  inclined  readings  to  horizontal  distances  informa- 
tion as  to  slope  of  surface  is  obtained  while  the  measur- 
ing is  being  done.  Distances  are  readily  obtained  with  the 
stadia  but  cannot  beset  off,  as  with  a  tape  or  chain. 

Stadia  is  a  word  with  the  same  root  as  "stadium"  the 
Latin  form  of  "Stadion,"  the  principal  Greek  measure  of 
length,  a  little  less  than  an  eighth  of  a  mile.  Stadia  rods  are 
often  called  "telemeter"  ("afar-off  measuring")  rods.  In 
Europe  a  theodolite,  designed  to  be  used  principally  for 
topographical  work  and  equipped  with  gradienter,  stadia 
wires,  etc.,  is  called  a  "tachymeter,"  that  is,  "a  rapid 
measurer." 


TRANSIT  SURVEYING 
STADIA  REDUCTION  TABLE 


251 


M. 

0° 

1° 

2° 

ro'.. 

i  2' 

Hor. 

dist. 

IOO.OO 
IOO   OO 

Diff. 
Elev. 

o.oo 
o  06 

Hor. 
dist. 

99-97 
QQ  Q7 

Diff. 
elev. 

1.74 
I   80 

HOT. 

dist. 

99.88 

QQ     87 

Diff. 
elev. 

3-49 

3r  ir 

A 

IOO   OO 

O    12 

QQ     Q7 

I  86 

QQ   87 

3    60 

6  

IOO   OO 

O    17 

QQ  q6 

I    Q2 

QQ     87 

3  66 

8  

IOO   OO 

O.  23 

QQ   Q6 

I    98 

QQ   86 

•2    72 

10  

IOO.OO 

O.  2Q 

QQ.Q6 

2  .04 

99  86 

3    78 

12  

IOO   OO 

o  3< 

QQ   Q6 

2    OQ 

QQ   81; 

3    84    % 

14  

IOO   OO 

o  4.1 

QQ     QC 

2    1C 

QQ  8s 

3QO 

16  

IOO   OO 

O.47 

QQ    QC 

2    21 

QQ   84 

•1     Qf 

18  

IOO.OO 

O.  <\2 

QQ.QC 

2  .  27 

QQ   84 

4  OI 

20 

IOO   OO 

o  ^8 

QQ     Qf 

2     73 

QQ   8  3 

4O7 

22  

IOO   OO 

o  64 

QQ  Q4 

2    38 

QQ   8l 

41  3 

24  

IOO   OO 

o  70 

QQ   Q4 

2  44 

i7^-"O 

QQ     82 

4   l8 

26  

no   no 

o.  76 

QQ  Q4 

2  .  SO 

QQ   82 

4    24 

28 

QQ   QQ 

o  81 

QQ     Q3 

2    ?6 

QQ   8  1 

47Q 

30  
32.  . 

99-99 
QQ  QQ 

0.87 

O  Q3 

99-93 

QQ  Q3 

2.62 

2    67 

99-81 
QQ  80 

4.36 

4   42 

34  
36  
38  

4O 

99-99 
99-99 
99-99 

QQ     QQ 

0.99 
I-OS 
I.  II 

i  16 

99-93 
99-92 
99-92 

QQ     Q2 

2.73 

2-79 

2.85 

2    QI 

99-80 

99-79 
99-79 

QQ     78 

4.48 

4-53 
4-59 
46c 

42 

QQ     QQ 

22 

QQ    QI 

2Q7 

QQ    78 

471 

44  
46  

48 

99.98 
99.98 

QQ   Q8 

.28 

•34 
4.0 

99.91 
99.90 
QQ    QO 

3-02 

3.08 

314 

99-77 
99-77 

QQ    76 

4.76 
4.82 

4  88 

eo.  . 

QQ   Q8 

41? 

QQ   QO 

3    2O 

QQ     76 

4Q4 

<?2 

QQ     Q8 

(?T 

QQ     8Q 

3    26 

QQ    7t 

4QQ 

"?4 

QQ   Q8 

(?7 

QQ    8Q 

3-2T 

QQ      74 

5QC 

*3  
e6 

QQ     Q7 

63 

QQ     8Q 

3-37 

QQ    74 

5      II 

58 

QQ     Q7 

69 

QQ     88 

3A-1 

QQ    73 

r    17 

60  

QQ   Q7 

74 

yy  •"«-> 
QQ   88 

-}     AQ 

QQ    72 

e    23 

5=o.75 

0-75 

O.OI 

0-75 

O.O2 

0-75 

0.03 

C=i.oo 

i  .00 

O.OI 

I  .OO 

0.03 

I  .OO 

O.O4 

C=I.2S 

1-25 

O.O2 

1.25 

0.03 

1-25 

0.05 

252 


PRACTICAL  SURVEYING 
STADIA  REDUCTION  TABLE  (Continued) 


M. 

3° 

4° 

5° 

o'  

HOT. 

dist. 

00.72 

Diff. 
elev. 

<  .  23 

HOT. 
dist. 

qq.  Si 

Diff. 
elev. 

6  96 

HOT. 
dist. 

qq   24 

Diff. 
elev. 

8  68 

2  

QQ.72 

q.28 

qq.  51 

7  .02 

qq  23 

8   74 

A 

QQ   71 

tf       -2A 

QQ    <?O 

7   Q7 

QQ      22 

8  80 

6 

QQ   71 

^      4O 

QQ   4Q 

7    13 

QQ    21 

8  85 

8          

QQ.  7O 

^      46 

qq  48 

7    IQ 

QQ     2O 

8  QI 

10  

qq.6q 

<\  .  ^2 

QQ  47 

7   2< 

qq  iq 

8  q? 

12 

QQ   6Q 

e    r7 

qq  46 

7    3O 

qq  1  8 

902 

14. 

qq  68 

5  62 

qq  46 

7   ^6 

qq  17 

q  08 

16 

qq  68 

5  69 

QQ  4^ 

7   42 

qq  16 

Q    14 

18         

qq  67 

c   7C 

QQ  44 

7  48 

QQ    je 

920 

20     

99-66 

5  80 

QQ  43 

7   "?3 

qq  14 

92C 

22 

qq  66 

S  86 

QQ   42 

7    S9 

qq  13 

q  31 

24. 

QQ  6<? 

^   Q2 

QQ   41 

/  oy 
7  6<; 

qq  ii 

q  37 

26              .... 

QQ  64 

e  q8 

QQ  4O 

7   71 

qq  10 

943 

28         .... 

qq  63 

6.04 

QQ.  3Q 

7.76 

qq  oq 

••to 

Q     48 

3O 

qq.63 

6.oq 

qq.  ^8 

7.82 

qq  08 

Q    ^4 

32  .  . 

99.62 

6.15 

99.38 

7.88 

qq.O7 

q.6o 

24. 

qq  62 

6    21 

qq  37 

7  q4 

qq  06 

q  6s 

II::::::::.: 
3»....  

4O 

99  -61 
99-6o 
qq.  eq 

6.27 

6-33 
6.38 

99-36 
99-35 
qq.34 

7-99 
8.05 
8  ii 

99-05 
99-04 
qq  03 

9.71 

9-77 
q  83 

42  
44  

99-59 
99.58 

6.44 
6.50 

99-33 
99.32 

8-17 

8.22 

99-oi 
99.00 

9-88 

q.q4 

46  
48  
50....  ...... 

^2 

99-57 
99.56 
99-56 

qq   "i1? 

6.56 
6.61 
6.67 

6  73 

99-31 
99-30 
99-29 

QQ    28 

8.28 

8-34 
8.40 

8  4<? 

98.99 
98.98 
98.97 

q8  q6 

10.00 

10.05 

IO.II 
IO    1  7 

tA 

qq   ">4 

6  78 

QQ   27 

8  51 

q8  q4 

IO    22 

It.:::.:::: 

"?8 

99-53 

qq.  52 

6.84 
6.qo 

99-26 

qq.  2< 

8-57 
8.63 

98.93 
q8  q2 

10.28 
IO    34 

60  

qq.  ci 

6.96 

QQ.  24 

8.68 

q8  qi 

10.40 

C=o.75 

0-75 

0.05 

0.75 

0.06 

0.75 

O.O7 

C=i.oo 

I.OO 

0.06 

I.OO 

0.08 

0.99 

0.09 

C=i.25 

1.25 

0.08 

1.25 

O.IO 

1.24 

O.  II 

TRANSIT  SURVEYING 

STADIA  REDUCTION  TABLE  (Continued) 


253 


M. 

6° 

7° 

3° 

o'  

HOT. 
dist. 

q8  qi 

Diff. 
elev. 

10  40 

Hor. 
dist. 

qg   ci 

Diff. 
elev. 

12    IO 

HOT. 
dist. 

98  06 

Diff. 
elev. 

13    78 

2  

Q8    QO 

10  4C 

yu-  o* 
q8  co 

12    1C 

qg  oc 

13    84 

4  

98.88 

IO   C-i 

98  48 

12    21 

qg  03 

13  8q 

6  

98.87 

IO.  C7 

98  47 

12    26 

y^-^o 
q8  01 

13  CK 

8  

98.86 

10.62 

98.46 

12.32 

•J     v 
98  oo 

UOI 

10  
12  

98.85 
98.83 

10.68 
10.  74 

98.44 
98  43 

12.38 
12   43 

97-98 
97   97 

14.06 

UI? 

14  

16  
18  
20  

22  

98.82 
98.81 
98.80 
98.78 

98.77 

10.79 
10.85 
10.91 
10.96 

ii  .02 

98.41 
98.40 
98-39 
98-37 

98    36 

12.49 

12.55 
I  2.  60 

12.66 

1272 

97-95 
97-93 
97-92 
97-90 

97  88 

14.17 
14.23 
14.28 
14.34 

14   4O 

24  

98.76 

n.o8 

98.34 

12  .  77 

97   87 

U4C 

26 

q8   74 

ii  i  3 

qg  -}•> 

12    83 

q7   8c 

14    CI 

28  
30  

32  .  . 

98.73 
98.72 

Q8.7I 

11.19 

11-253 

ii  ^o 

98.31 
98.29 

98  28 

12.88 

12.94 

13   OO 

97.83 
97.82 

Q7   80 

14.56 
14.62 

U6? 

34  

98.69 

ii  .  36 

98.27 

13  .OC 

y/  «w 

97    78 

14   73 

36  

98.68 

ii  .42 

98.21? 

13.11 

97    ?6 

U79 

?8 

98  67 

II   47 

qg   24 

13    17 

Q7    7C 

U84 

40  
42  

98.65 
08.64 

n-53 
II  .  cq 

98.22 
98   2O 

13.22 
13    28 

97-73 

Q7    71 

14.90 
14  QC 

44  

98.63 

II  .64 

98  iq 

1333 

97   6q 

1C   OI 

46 

q8  61 

1  1    7O 

Q8    17 

1  3    3Q 

q?  68 

\o  •'-'* 
1C   06 

48  
50  

C2.  . 

98.60 
98.58 

98.  C7 

11.76 
11.81 

11.87 

98.16 
98.14 

Q8   13 

13-45 
I3-50 

13    ^6 

97-66 
97-64 

q?  62 

15-12 
I5.I7 

1C    23 

54  

98.56 

II  .Q3 

q8  ii 

13  61 

y/  •«•« 
97  61 

IS    28 

56  
58 

98.54 
q8  C3 

11.98 

12    O4 

98.10 
qg  08 

13.67 

13    73 

97-59 

Q7    C7 

15-34 
I  C    4O 

60  

98  c  i 

12    IO 

q8  06 

13    78 

Q7    CC 

1C    4C 

C=o.7S 

0.75 

0.08 

0-74 

O.IO 

0.74 

O.II 

C=i  .00 

0-99 

O.II 

o-99 

0.13 

o-99 

0.15 

C=i.25 

1.24 

0.14 

1.24 

o.  16 

1.23 

0.18 

254 


PRACTICAL  SURVEYING 
STADIA  REDUCTION  TABLE     (Continued} 


M. 

9° 

10° 

n° 

0' 

Hor. 
dist. 

Q7    I  C. 

Diff. 
elev. 

T  c    A  e 

Hor. 
dist. 

06  08 

Diff. 
elev. 

Hor. 
dist. 

06  26 

Diff. 

elev. 

18   72 

2  

Q7  •  13 

ICCT 

Q6   Q6 

17    16 

yu-o" 
q6  74. 

10-  16 

18  78 

4 

Q7    12 

i  i  16 

Q6    QA 

17    21 

y^-^4 

Q6    32 

1  8  84 

6  
8  
10           

97-50 
97.48 
Q7   4.6 

15.62 
15.67 

jr    77 

9<3.92 
96.90 

96  88 

17.  26 
17.32 
17    37 

96.29 
96.27 
Q6    2^ 

18.89 
i8.95 

IQ    OO 

12  

07.44 

ic   78 

96  86 

17   43 

Q6    23 

IQ    O< 

14. 

Q7   4.3 

K  84 

06  84. 

17   4.8 

yu.z^ 

q5  21 

19    1  1 

16  
18  
20           .... 

97-41 

97-39 

Q7    37 

15-89 
15-95 
16  oo 

96.82 
96.80 
Q6  78 

17-54 
17-59 
17  6c. 

96.18 
96.16 

Q6    14. 

19.16 
19.21 
IQ    27 

22  

07    2C 

16  06 

Q6  76 

17    7O 

96    12 

IQ    32 

24  
26  
28  
3O    . 

97-33 
97-31 
97-29 
Q7    28 

16.11 
16.17 
16.22 
16  28 

96.74 
96.72 
96.70 

96  68 

17.76 
17.81 
17.86 
17   Q2 

96.09 
96.07 
96.05 
Q6   O3 

19-38 
19-43 
19.48 
IQ    14. 

32  .  . 

Q7    26 

16  33 

96  66 

17   Q7 

06  oo 

IQ    ^Q 

34-  • 

Q7   24. 

16.  3Q 

0.6.64 

18  03 

Q1^    Q8 

IQ    64 

36  
38  
40  

42  

97-22 

97-20 
97.18 

Q7   l6 

16.44 
16.50 
16.55 

16  61 

96.62 
96.60 
96.57 

Q6      << 

18.08 
18.14 
18.19 

18   24 

95.96 

95-93 
95-91 

qc  8q 

19.70 

19-75 
I9-80 

IQ  86 

44  <  .  .  .  . 

Q7    14 

16  66 

Q6      ^ 

18  30 

95  86 

IQ   QI 

46  
48  
So  

•>2.  . 

97.12 
97.10 
97-08 

Q7   06 

16.72 
16.77 
16.83 

16  88 

96.51 

96.49 
96.47 

q6  41 

18.35 
18.41 
18.46 

18  <i 

95.84 
95.82 

95-79 
qc   77 

•*  ~J 

I9-96 
2O.O2 
20.07 

20.  12 

C.4. 

Q7   O4. 

1  6    QJ. 

Q6   4.2 

18  ^7 

qe    7  e 

20  18 

\l:::::..:.. 

$8 

97-02 

Q7    OO 

16.99 

17   OC. 

96.40 

06  a8 

18.62 

18  68 

95-72 

CK  70 

20.23 
20  28 

60  

06  08 

17    IO 

57V'<9" 

06  ^6 

18  73 

qc  68 

2O   34 

C=o.75 

0.74 

O.I2 

0.74 

0.14 

0-73 

0.15 

C=i.oo 

0-99 

0.16 

0.98 

0.18 

0.98 

O.2O 

C=1.25 

1.23 

O.2I 

1.23 

0.23 

I  .  22 

0.25 

TRANSIT  SURVEYING 
STADIA  REDUCTION  TABLE     (Continued) 


255 


M. 

12° 

13° 

14° 

o' 

Hor. 
dist. 

OC  68 

Diff. 
elev. 

20   34 

Hor. 
dist. 

Q4-94 

Diff. 
elev. 

21  .02 

Hor. 

dist. 

04.  ic 

Diff. 
elev. 

23  47 

2 

CK    6< 

2O    3Q 

94-  Qi 

21  .07 

04-12 

23  .  "\2 

4 

CK    63 

20.44 

94.89 

22.O2 

04.00 

23.58 

6  
8   

95.61 

Q5.58 

20.50 
20.55 

94-86 
94.84 

22.08 
22.13 

94-07 
94.04 

23-63 
23-68 

10  

•yj    o 

gc.  c6 

20.60 

94.81 

22.  IS 

94.01 

23.73 

12                .     . 

qe    c-2 

20.66 

94-  7Q 

22.23 

93.98 

23.78 

14 

qc  .  ci 

20.  71 

04.76 

22.28 

93-95 

23-83 

16   

qc  .4q 

20.76 

94-73 

22.34 

93.93 

23.88 

18  

015.46 

20.81 

94.71 

22.39 

93.90 

23.93 

20  

qc.44 

20.87 

94.68 

22.44 

93.87 

23.99 

22     

QC  41 

2O.  Q  2 

04.66 

22.49 

93.84 

24.04 

24       

qc   2Q 

2O.  Q7 

94.63 

22.54 

03.81 

24.00 

26  

qc.a6 

21  .03 

•yt      o 
04.6o 

22.6O 

93-79 

24.14 

28 

qc    74. 

21    08 

04    ^8 

22    6< 

03  76 

24   IO 

2Q 

qr    52 

21    13 

QA    cr 

22  .  70 

03   73 

24    24 

32  . 

0"?    2Q 

21  .l8 

04.  52 

22.7"? 

03-7O 

24.  2Q 

74 

05    27 

21  .  24 

94.  5O 

22.80 

yo    ' 
93.67 

24.  34 

36      - 

qc.24 

21  .  2q 

94-47 

22.85 

93-  6< 

24.39 

38.    . 

QC.22 

21  .  34 

94  44 

22.91 

03.62 

24  .44 

4O  

CK.IQ 

21  .39 

94.42 

22.96 

93-59 

24.49 

42  

44          ... 

95-17 

qc   14 

21.45 

21  .  ^O 

94-39 
04-  36 

23.01 
23.06 

93-56 

Q2  .  C7 

24-55 
24.60 

46  
48  

95-12 
Q">.oq 

21-55 
21.60 

94-34 
94-  31 

23.11 
23.16 

93-So 
93-47 

24-65 
24.70 

co.  . 

95-O7 

21.66 

94.28 

23.22 

93-45 

24.  75 

*\2  .  . 

(K  04 

21  .  71 

Q4.  26 

23  .  27 

03.42 

24.80 

«54.  . 

Qf?  .02 

21  .  76 

04.  23 

23.32 

93  •  39 

24.85 

56 

Q4   QQ 

21    8l 

04   2O 

f-2      77 

03    36 

24   QO 

58  

Q4   Q7 

21    8? 

04  17 

23    42 

03    33 

24   QC 

60  

Q4  Q4 

21    Q2 

04  IS 

23   47 

03  .  3O 

2<  .OO 

C=o.7S 

0-73 

0.16 

0.73 

0.17 

0-73 

O.I9 

C-i.oo 

0.98 

O.  22 

0-97 

0.23 

0.97 

0.25 

C=I.2S 

I  .22 

0.27 

I  .  21 

O.2O 

I  .21 

0.31 

256 


PRACTICAL  SURVEYING 
STADIA  REDUCTION  TABLE     (Continued) 


M. 

15° 

16° 

17° 

o'  

2          

Hor. 

dist. 

93-30 
Q3  .  27 

Diff. 
elev. 

25.00 
2^    OS 

Hor. 
dist. 

92.40 
Q2    37 

Diff. 
elev. 

26.50 
26    ^  ^ 

Hor. 

dist. 

91-45 
QI    42 

Diff. 
elev. 

27.96 
28  01 

4   

Q3  •  24 

2S    IO 

Q2    34 

•«  •  oo 

26    ^Q 

QI     3Q 

28  06 

6  

Q3  •  21 

2<  .  I"> 

Q2    31 

26    64 

QI    3  ^ 

28  10 

8  

93.18 

25  •  2O 

Q2.  28 

26    69 

QI    32 

28  15 

10 

Q2     l6 

2C    2S 

Q2    2  S 

26    74. 

QI    2Q 

12  

93-  13 

25    3O 

Q2    22 

26    7Q 

91    26 

28  25 

14  

Q3  •  10 

2^.31; 

Q2    IQ 

26  84 

QI    22 

28  30 

16  

93.07 

25.40 

Q2    I  < 

26.89 

QI    IQ 

28    34 

18 

07     O4 

2">    4<? 

Q2    1  2 

26    Q4. 

QI    l6 

«  '^ 
28    3Q 

20  
22  

93-Qi 

Q2.Q8 

25-50 
2<J.  « 

92.09 
92    06 

*;-3f* 

26.99 

27    OA 

91  .  12 
QI    OQ 

28.44 

28    dQ 

24  

92.01; 

25.60 

Q2  .03 

27    OQ 

91    06 

28    s4 

26  

92.92 

25.65 

Q2  .OO 

27  .  1^ 

QI    O2 

28    & 

28  
30  

32  .  . 

92.89 
92.86 

92.8^ 

25.70 

25-75 

25.80 

91-97 
91-93 

QI    QO 

27.18 
27-23 

27.28 

90.99 
90.96 

QO   Q2 

28.63 
28.68 

28    73 

34.  . 

•Q2.8o 

25.85 

91.87 

27  .  33 

QO  SQ 

28    77 

2 
36.  . 

92.77 

25.90 

QI  .84 

27.38 

QO  86 

28   82 

38  
40  

42  

44  
46  
48  
TO 

92-74 
92.71 

92.68 
92.65 
92.62 

92.59 
Q2  .  c6 

25-95 
26.0O 

26.05 
26.10 
26.15 
26.  2O 
26.  2< 

9I.8I 
91-77 

91-74 

9x-7i 

91.68 

9I-65 
QI    6l 

27-43 
27.48 

27.52 
27-57 
27.62 

27.67 

27    72 

yw.u^ 

90.82 
90.79 

90.76 
90.72 
90.69 
90.66 

GO    62 

28.87 
28.92 

28.96 
29.01 
29.06 
29.11 
2Q    I  ^ 

52  .  . 

92.53 

26.30 

QI.  ^8 

27.77 

QO    "CQ 

2Q    2O 

"\4 

Q2   4Q 

26    3< 

QI    ^s 

27    8l 

QO    ^  tj 

2Q    2  ^ 

It::::::::: 

58. 

92.46 
Q2   4^ 

26.40 
26.41? 

91.52 

91  48 

27.86 

27   QI 

90.52 
QO   48 

29  •  30 
2Q    34 

60  

Q2  .40 

26.  50 

QI    4$ 

27    96 

QO  4.< 

2Q    3Q 

C=o.7S 

0.72 

O.  2O 

0.72 

O.2I 

0.72 

0.23 

C=i.oo 

0.96 

0.27 

0.96 

0.28 

0-95 

0.30 

C=1.25 

1.20 

0-34 

1  .  2O 

0.36 

I.I9 

0.38 

TRANSIT  SURVEYING 
STADIA  REDUCTION  TABLE  (Continued) 


257 


M. 

18° 

19° 

J 

•0° 

o'  

HOT. 
dist. 

QO.45 

Diff. 
elev. 

2Q.  3Q 

HOT. 

dist. 

SQ  .40 

Diff. 
elev. 

30.78 

HOT. 

dist. 

88.30 

Diff. 
elev. 

32   14 

2 

QO  42 

2Q    44 

SQ  36 

3O  83 

88  26 

32    l8 

yw.^ 

QO    38 

*y  -H-t 
2Q    48 

<_>y  .  jv 
SQ  33 

o'-'-"*) 
3O  8? 

88  23 

32    23 

6 

QO   3S 

2Q    ^3 

SQ  2Q 

3O   Q  2 

88  19 

32    27 

8          

QO.  31 

2Q.58 

89  26 

3O   Q7 

88  15 

32    32 

IO               .... 

00.28 

2Q  .62 

89.22 

31    OI 

88  ii 

32    36 

12  
14 

90.24 

QO    21 

29.67 

2Q  .  72 

89.18 
SQ  15 

31.06 
31    IO 

88.08 
88  04 

32.41 
72    AC 

16 

90.l8 

2Q  .  76 

rSJ'*3 

SQ  ii 

31    I"? 

88  oo 

32    4Q 

18           .    ..  . 

QO.  14 

2Q.8l 

SQ  08 

31    IQ 

87  Q6 

32    CJ4 

20  

22  
24               .... 

90.11 
90.07 

QO   O4 

29.86 

29.90 

2Q    Q5 

89.04 

89.00 
88  96 

3!-24 

31.28 
31    33 

87.93 

87.89 
87  85 

32.58 
32.63 

32    6? 

26          

QO   OO 

3O.OO 

88   Q3 

31    38 

87  81 

32    72 

28    

8Q.Q7 

3O.O4 

88  89 

31    42 

87    77 

32    76 

3O.  . 

8Q.Q3 

3O.  OQ 

88.86 

31    47 

87.74 

32.80 

72 

SQ  QO 

3O    14 

88  82 

31    51 

87    7O 

32    85 

34 

89  86 

3O    IQ 

88  78 

0A  Ox 

31    56 

87  66 

32    8Q 

36 

89.83 

3O.  23 

88  75 

31    60 

87  62 

32    Q3 

38.. 

8Q.7Q 

30.28 

88  71 

31   65 

87  58 

32   Q8 

4O  

89.76 

^O.    ^2 

88  67 

31    6Q 

87.54 

33   O2 

42 

SQ  72 

3O    37 

88  64 

31    74 

87    51 

33   O7 

44 

89  69 

3O    41 

88  60 

31    78 

87   47 

33    II 

46  

89.65 

3O   46 

88  56 

31    83 

87   43 

3315 

48  

89.61 

^O.  ^1 

88  53 

31    87 

87    3Q 

33    2O 

SO.  . 

89.58 

30.  5^ 

88  4Q 

31    Q2 

87    35 

33    24 

$2  .  . 

SQ  $A 

30  60 

88  4<; 

31    Q6 

87    31 

33    28 

^4-  - 

SQ  51 

30  6^ 

88  41 

32   OI 

W  -OA 
87    27 

33    33 

56.. 

8Q.47 

30  60 

88  38 

32    O5 

87    24 

33    37 

f  

60  

89.44 
89.40 

30.74 
30.78 

88.34 
88.30 

32.09 
32.14 

87.20 
87.16 

33-41 
33-46 

£=0.75 

0.71 

0.24 

0.71 

0.25 

0.70 

0.26 

C=i  .00 

0-95 

0.32 

0-94 

0-33 

0-94 

0-35 

C-I.25 

1.19 

0.40 

1.18 

0.42 

I.I7 

0.44 

PRACTICAL  SURVEYING 

STADIA  REDUCTION  TABLE  (Continued) 


M. 

21° 

22° 

23° 

o' 

Hor. 

dist. 

87  16 

Diff. 
elev. 

33  .46 

Hor. 
dist. 

8<?.Q7 

Diff. 
elev. 

34-  73 

Hor. 

dist. 

84   73 

Diff. 
elev. 

2C     07 

2 

87.12 

33.  SO 

8<?.Q3 

34-  77 

84.69 

36  .01 

87.08 

33.54 

85.89 

34.82 

84.65 

°- 
36.05 

6          .    ..  . 

87.04 

33-59 

85-85 

34-86 

84.61 

36.09 

8         

87.00 

33.63 

85.80 

34.90 

84.57 

36.13 

10       

86.96 

33.67 

85.76 

34-94 

84.52 

36.17 

12                   .... 

86.Q2 

33.72 

85.72 

34.98 

84.48 

36.  21 

14. 

86.88 

33.76 

85.68 

35.02 

84.44 

36.25 

16 

86  84 

33   80 

85  64 

3^  .07 

84  40 

36    2Q 

18 

86  80 

33.84 

85.60 

35.11 

84   3^ 

36.  33 

20 

86.77 

33.8Q 

85.56 

35.15 

84.31 

36.  37 

22               

86.73 

33.93 

85.52 

35.19 

84.27 

36.41 

24. 

86.69 

33.97 

85.48 

35.23 

84.23 

36.45 

26    

86.65 

34.01 

85.44 

35-27 

84.18 

36.49 

28 

86.  61 

34.00 

85.40 

35.31 

84.  14 

r 

30.  53 

•2Q 

86.57 

34.10 

85.36 

35.36 

84  .  10  • 

36.57 

32 

86.53 

34.14 

85.31 

35.40 

84.06 

36.6l 

34    .            .... 

86.49 

34-i8 

85.27 

35-44 

84.01 

36.65 

36  

86.45 

34-23 

85-23 

35-48 

83-97 

36.69 

38 

86.41 

34.27 

85.19 

35.52 

83-93 

36.73 

40 

86.37 

34.31 

85.15 

35.56 

83.89 

36.77 

42         

86.33 

34-35 

85.11 

35  -60 

83.84 

36.80 

44    

86.29 

34-40 

85.07 

35  -64 

83-80 

36.84 

46 

86.25 

34-44 

85.02 

35-68 

83.76 

36.88 

48 

86.21 

34-48 

84.98 

35-72 

83.72 

36.92 

"?O 

86.17 

34.52 

84.94 

35-76 

83.67 

36.96 

^2 

86  13 

34-  57 

84.90 

35.80 

83-63 

37-oo 

^4 

86.  oq 

Ot     Jl 

34.61 

84.86 

35-85 

83-59 

37-04 

s6 

86.05 

34.65 

84.82 

35-89 

83-54 

37/o8 

<;8 

86.01 

34.69 

84.77 

35-93 

83.50 

37.12 

60"      

85.97 

34-73 

84.73 

35-97 

83.46 

37.16 

C=o.75 

0.70 

0.27 

0.69 

0.29 

0.69 

0.30 

C=i.oo 

0-93 

0-37 

0.92 

0.38 

0.92 

0.40 

C=I.2S 

1.16 

0.46 

1.1$ 

0.48 

I-I5 

0.50 

TRANSIT  SURVEYING 
STADIA  REDUCTION  TABLE  (Continued) 


259 


M. 

24° 

25° 

26° 

o'    .. 

Hot. 
dist. 

83  46 

Diff. 
elev. 

37  .  16 

.    HOT. 
dist. 

82   14 

Diff. 
elev. 

38   30 

HOT. 

dist. 

80  78 

Diff. 
elev. 

3Q  4O 

2     .          ... 

83.41 

37  .  20 

82  09 

38    34 

80   74 

3Q  44 

4   

8^.17 

37-23 

82.  o<; 

38  38 

80.69 

3Q.47 

6  

83.33 

37.27 

82.01 

38.41 

80.65 

30  .  ci 

8  

83.28 

37.31 

81.96 

38.  4"? 

80.60 

30.  C4 

10  

83.24 

37-35 

81.92 

38.49 

8o.<;<; 

39.";8 

12     

83.20 

37  •  3Q 

81  87 

38    S3 

80  <?i 

3Q  6l 

14  

16  

83-iS 
83.11 

37-43 
37-47 

81.83 
81.78 

38.56 
38.60 

80.46 
80.41 

39-65 

30.60 

18  
20  

22  

83.07 
83.02 

82.98 

37-51 
37-54 

37.58 

81.74 
81.69 

81.65 

38.64 
38.67 

38   71 

80.37 
80.32 

80  28 

39.72 
39-76 

70.70 

24  

82.03 

37.62 

81.60 

38.7^ 

80.23 

39.83 

26  

82.89 

37-66 

81.56 

38.78 

80.  18 

39-86 

28...'  

7Q 

82.85 
82  80 

37-70 
37   74 

81.51 

8l    4.7 

38.82 
38  86 

80.14 

80   OQ 

39-90 
3Q  Q3 

32.   . 

82.76 

37-77 

81.42 

38  89 

80.04 

30.07 

34.  . 

82.72 

37.81 

81.38 

38.03 

SO.OO 

40100 

36  
38  
40  

42  

82.67 
82.63 
82.58 

82.54 

37-85 
37.89 
37-93 

37.96 

81.33 
81.28 
81.24 

81  19 

38.97 
39-00 
39-04 

3Q.o8 

79-95 
79-90 

79-86 
7Q.8l 

40.04 
40.07 
4O.II 

40.  14 

44  

82.49 

38.00 

81.1? 

^Q.  II 

70.76 

40.  18 

46  
48 

82.45 
82  41 

38.04 
38  08 

81.10 
81  06 

39-15 

3Q    l8 

79.72 
7Q   6? 

40.21 

4O    24 

eo.  . 

82.36 

38.11 

81  01 

30    22 

7Q   62 

4.0   28 

52  
54  
56 

82.32 
82.27 
82.  23 

38.15 
38.19 

38    23 

80.97 
80.92 
80  87 

39-26 

39-29 
30    33 

79-58 

79-53 

7Q  48 

40.31 
40-35 
40    38 

58.  . 

82.18 

38.26 

80  83 

3Q    36 

7Q   44 

4.O      42 

60  -.. 

82.14 

38.30 

80  78 

3Q   4O 

7Q    3Q 

AO  45 

£=0.75 

0.68 

0.31 

0.68 

0.32 

0.67 

0.33 

C=i.oo 

0.91 

0.41 

0.90 

0-43 

0.89 

0.45 

C=I.2S 

1.14 

0.52 

1-13 

0-54 

I  .12 

0.56 

260 


PRACTICAL  SURVEYING 
STADIA  REDUCTION  TABLE 


M. 

27° 

28° 

29° 

30° 

o'  

Hor.       Difif. 
dist.       elev. 

79-39     4°  -45 

Hor.         Diff. 
dist.         elev. 

77.96     41.45 

Hor.         Diff. 
dist.         elev. 

76.50     42.40 

Hor.         Diff. 
dist.         elev. 

75.00     43.30 

2  

4  

79.34     40.49 
79.30     40.52 

77.91      41.48 
77.86     41.52 

76.45      42.43 
76.40     42.46 

74  -  95     43  •  33 
74.90     43.36 

6 

7Q  2  ^     40  .  ^  t; 

77  .81     41  .  55 

76.3?      42.40 

74  8  <?     43    3Q 

8 

7Q  .20      4O  .  ZQ 

77.77     41.58 

76.30     42.53 

74.80     43  42 

10  

12 

79.15      40.62 

79  ii     40  66 

77.72     41.61 
77.67     41.65 

76.25      42.56 
76.20     42  59 

74-75     43-45 
74  70     43  47 

14 

79  .  06    40  69 

77.62     41  .68 

76.15     42.62 

74  6<     43   550 

16 

79.01     40.72 

77.57     41.71 

76.10    42.65 

74.60     43  .  53 

18  
20  

78.96    40.76 
78.92     40.79 

77.52     41.74 
77.48     41.77 

76.05     42.68 
76.00     42.71 

74-55     43-56 
74.49     43.59 

22  

24  
26  
28  
3O.  . 

78.87     40.82 
78.82     40.86 
78.77     40.89 
78.73     40.92 
78  .68    40  .  96 

77.42     41.81 
77.38     41-84 
77,33     41.87 
77.28     41-90 
77.23     41.93 

75-95     42.74 
75.90    42.77 
75.85     42.80 
75.80    42.83 
75-75     42.86 

74.44    43.62 

74-39     43-65 
74.34    43.67 
74  .29     43  .  70 
74  .24     43  .  73 

32   . 

78.63      4O.  QQ 

77.18     41.97 

75.70    42.89 

74.19     43.76 

34  
36..  . 

78.58      41.02 
78.54      41.06 

77.13     42.00 
77.09     42.03 

75.65     42.92 
75.60    42.95 

74.14    43.79 
74.09     43.82 

38 

78.49      41.09 

77.04    42.06 

75-55     42.98 

74  .04    43  .  84 

40  

42  
44  
46  
48  
So  

52.  . 

78.44      41.12 

78.39      41.16 
78.34      41.19 
78.30      41.22 
78.25      41.26 
78.2O      41.29 

78.15      41.32 

76.99     42.09 

76.94     42.12 
76.89     42.15 
76.84    42.19 
76.79     42.22 
76.74    42.25 

76.69     42.28 

75.50    43.01 

75.45     43-04 
75.40    43.07 
75-35     43-iQ 
75.30    43.13 
75.25     43.16 

75.20    43.18 

73-99     43-87 

73-93     43-90 
73.88     43.93 
73-83     43-95 
73-78     43.98 
73.73     44.01 

73  .68     44  .  04 

$: 

78.10      41.35 

78  06    41  39 

76.64    42.31 
76.  <Q      42  .  34 

75.15     43.21 
75  .10    43  .  24 

73.63     44.07 
73.58     44.09 

58. 

78.01     41  42 

76.55      42.37 

75.05     43.27 

73.52     44.12 

60  

77.96    41.45 

76.50     42.40 

75.00    43.30 

73.47     44.15 

C=o.75 

0.66       0.35 

0.66       0.36 

0.65      0.37 

0.65       0.38 

C=i.oo 

o  .  89      o  .  46 

0.88      0.48 

0.87      0.49 

0.86      0.51 

C=i.25 

I.  I  I      0.58 

i.io      0.60 

1.09      0.62 

i  .  08      o  .  64 

TRANSIT  SURVEYING  261 

The  subdividing  of  land  into  building  lots;  the  re-survey 
of  lots;  keeping  records;  filing  notes  and  records;  setting 
monuments,  etc.,  are  all  fully  dealt  with  in  the  author's 
book  ''Engineering  Work  in  Towns  and  Cities."  ($3.00.) 

In  making  re-surveys  a  random  line  must  first  be  run; 
sometimes  several  such  lines.  If  the  compass,  and  chain 
were  employed  on  the  original  survey,  it  is  advisable  to 
run  the  random  lines  with  the  needle  and  employ  green 
hands  to  measure  with  an  old-fashioned  chain.  In  this  way 
the  locations  of  monuments  will  be  discovered  more  readily 
than  if  modern  methods  are  employed.  When  the  corners 
are  located  the  true  lines  should  be  run  with  the  transit 
and  a  steel  tape  used  by  skilled  helpers.  The  new  notes 
should  then  be  recorded. 

MAKING  THE  MAP 

A  complete  map  should  contain  all  the  data  necessary 
to  enable  a  competent  surveyor  to  re-trace  the  lines.  Not 
less  than  two  permanent  monuments  should  be  shown  on 
the  map,  connected  to  the  lines  of  the  survey,  with  tie 
lines  to  reference  points. 

The  following  items  should  be  shown  on  a  map  and  the 
list  is  given  as  a  sort  of  "specification  reminder"  for  the 
surveyor  and  draftsman. 

1.  Scale  of  map. 

2.  Meridian  line  with  declination  of  needle. 

3.  A  short  title  with  the  date  of  survey. 

4.  Monuments:  Location;  description;   tie  lines. 

5.  Lengths  of  all  lines. 

6.  Bearings  of  all  lines. 

7.  Angles  of  intersection  of  all  lines. 

8.  Name  of  owner  of  record. 

9.  Names  of  adjoining  owners  of  record  placed  on  their 
land  with  location  of  common  corners. 

10.  Names  of  all  recognized  landmarks  within  the  area 
embraced  in  the  map. 

11.  Statement  over  signature  of  the  surveyor  that  he 
has  carefully  checked  the  map,  and  that  it  truly  represents 
the  survey  made  by  him,  and  that  he  set  the  monuments 
described  and  drove  a  stake  at  each  corner. 


262  PRACTICAL   SURVEYING 

If  the  map  is  of  an  addition  to  a  village,  town  or  city, 
or  is  of  a  subdivision  of  a  tract  of  land  into  small  parcels 
or  lots,  it  should  contain  all  the  foregoing  and  in  addition 
the  following: 

12.  The  number  of  each  block  and  lot. 

13.  The  width  and  name  of  each  thoroughfare. 

14.  The   dedication,    acknowledged    before    a    qualified 
legal  officer,  of  the  public  thoroughfares  to  the  use  of  the 
public. 

Every  owner  whose  property  is  shown  on  the  map  as 
being  included  within  the  areas  dedicated  to  public  use 
must  sign  the  dedication. 

Mr.  S.  N.  Howard  of  Chicago,  in  the  Proceedings  of  the 
Illinois  Society  of  Engineers  and  Surveyors,  gave  the  fol- 
lowing list  of  items  to  be  shown  on  maps  of  city  lot 
surveys : 

"Lines,  grades,  angles,  location  and  elevation  of  sewer, 
elevation  of  walks  in  front  of  lot  and  adjoining  the  same 
and  elevation  of  ground;  and  sometimes  in  addition  is  the 
size  and  location  of  water,  gas  and  electric  mains;  the 
location  of  point  where  water,  gas  and  electric  service  pipes 
come  through  the  curb;  location  of  house  drains;  the 
location,  height  and  plumb  of  adjoining  buildings;  thick- 
ness of  party  walls  at  the  several  stories,  and  location  and 
depth  of  the  foundations  of  same." 

City  lot  surveys  are  usually  made  for  building  purposes 
and  an  architect  needs  all  the  above  data. 

Maps  of  transit  surveys  may  be  plotted  like  maps  of 
compass  surveys  with  protractors.  (See  Chapter  on  Com- 
pass Surveying.) 

When  for  any  reason  the  protractor  angles  are  not  ac- 
curate enough,  angles  may  be  set  out  to  the  nearest  minute 
by  using  either  a  table  of  chords  or  a  table  of  natural 
tangents.  If  the  surveyor  has  no  book  containing  a  table 
of  chords  he  can  use  natural  sines.  Twice  the  sine  of  half 
the  angle  is  the  chord  of  the  whole  angle. 

First  lay  off  a  base  ten  inches  (or  units)  long.  The 
tables  express  the  values  of  the  functions  as  decimal  frac- 
tions, so  this  length  of  10  =  unity  (i).  The  chord  (or 
natural  tangent)  is  measured  with  the  scale  used  for  the 
base. 


TRANSIT   SURVEYING  263 

In  Fig.  204  let  AC  =  base  =  unity  and  from  A  as  a 
center  describe  the  arc  BC.  Set  the  compass  to  scale  the 
chord  BC  of  the  angle  BA  C,  and  from  C  as  a  center  with 
radius  =  chord,  intersect  arc  BC  at  B.  Draw  a  straight 
line  from  A  through  B,  thus  forming  the  angle  BA  C. 


FIG.  204.  FIG.  205. 

In  Fig.  205  let  AC  =  base  =  unity,  and  at  C  erect  the 
perpendicular  BC.  On  the  perpendicular  lay  off  the  tan- 
gent of  the  angle  A.  The  line  is  then  drawn  from  A 
through  B  to  form  the  angle  BA  C. 

The  angles  may  be  set  off  from  the  ends  of  lines,  thus 
resembling  deflection  work,  or  two  lines  may  be  drawn 
normal  to  each  other  in  the  middle  of  the  sheet  and  a 
circle  drawn  with  radius  =  10  units  of  scale,  with  the  in- 
tersection as  the  center  of  the  circle.  Chords  may  be  set 
off  on  the  circumference  or  tangents  measured  from  the 
end  of  the  radius  for  all  the  angles.  The  lines  can  then 
be  transferred  by  using  triangles  and  straight-edge. 

If  the  error  in  the  survey  is  large  enough  to  show  on  the 
map  a  closure  must  be  "fudged,"  no  matter  how  accu- 
rately the  angles  are  drawn.  If  the  survey  is  balanced  so 
the  errors  are  distributed  it  will  be  necessary  to  calculate 
new  bearings  and  distances  if  angles  are  to  be  plotted. 
This  however  is  not  right,  besides  being  very  laborious. 
The  "latitude  and  departure  method"  enables  a  drafts- 
man to  make  an  accurate  map  on  which  the  lines  will  close 
and  it  is  used  by  all  experienced  men  who  are  noted  for 
careful  work. 

The  draftsman  must  remember  that  the  field  work  has 
been  done  with  all  possible  care.  In  spite  of  this  there  is 
an  error  of  closure,  the  amount  of  which  depends  on  the 
value  of  the  land.  The  error  is  probably  distributed 
throughout  all  the  courses  so  the  bearings  and  distances 
recorded  by  the  instrument  man  must  be  placed  on  the 
map.  The  error  of  closure  is  discovered  by  the  computa- 


264 


PRACTICAL  SURVEYING 


tions,  the  field  work  not  disclosing  it.     The  same  conditions 
must  appear  on  the  map  and  inaccuracies  in  plotting  added 

to  the  field  error  may  prevent 
the  lines  closing  by  a  quite  ap- 
preciable amount.  To  plat  the 
work  by  using  latitudes  and 
departures,  the  lines  must  close, 
and  furthermore  the  computa- 
tions  give  the  exact  amount  of 
&  error  and  the  surveyor  knows 
whether  it  lies  within  proper 
limits. 

The    computations    for    the 
field    shown    in   Fig.   206   are 
given  in  table  on  the  following  page. 

The  latitudes  and  departures  are  tabulated,  the  error 
found  and  distributed.  The  balanced  latitudes  and  de- 
partures are  then  algebraically  as  follows: 


FIG.  206. 


Latitudes 

-     599-51 
+    839.89 

+  240.38 
+  H28.58 
+  1668.96 
—  1059.22 
+  609.74 
+  347.89 
957.63 


~  1743.62 


Departures 

+  I273.96 
-j-     109.28 

+1383.24 

-  1075.26 

+     307.98 
—  1283.46 

-  975.48 

-  927.94 

-  1903.42 
+  927-66 


The  starting  point  has  no  latitude  or  departure.  In 
going  around  a  field  the  travel  north  equals  the  travel 
south  and  the  travel  east  equals  the  travel  west,  so  the 
algebraic  latitude  and  departure  for  the  last  station  must 
equal  the  balanced  latitude  and  departure  with  the  sign 
reversed.  This  checks  the  summation. 

The  method  of  plotting  is  shown  in  Fig.  207.  Through 
the  point  selected  for  Sta.  0  draw  the  horizontal  line  OX, 
OX'  and  the  vertical  line  OY,  OF'.  Divide  the  sheet  into 
squares  100  ft.  by  100  ft.  Number  them  as  shown  and  in 
each  square  in  which  a  corner  will  be  located  measure  the 


TRANSIT  SURVEYING 


265 


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Total 
latitude. 

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266 


PRACTICAL  SURVEYING 


fractional  parts,  draw  the  intersecting  latitude  and  depar- 
ture lines  and  locate  the  corner.  Finish  the  outline  of  the 
field  by  connecting  the  corners.  When  the  outlines  and  all 


FIG.  207.     Platting  by  latitudes  and  departures. 

bearings  and  lengths  are  inked  in,  the  co-ordinate  lines  are 
erased,  for  they  are  drawn  with  a  sharp,  hard  lead  pencil. 

Maps  expected  to  be  in  fairly  constant  service  (working 
maps)  should  be  drawn  on  the  best  quality  of  cloth  mounted 
paper  and  the  co-ordinate  lines  should  be  very  fine  red  ink 
lines.  These  maps  never  leave  the  surveyor's  office, 
tracings  being  made  of  such  portions  as  may  be  wanted 
by  clients.  In  many  cities  maps  are  drawn  on  sheets  with 
co-ordinate  lines  drawn  in  ink.  The  co-ordinates  (latitude 
and  departure)  are  marked  in  figures  at  each  monument 
and  lot  corner.  The  bearing  and  distance  between  any 
two  points  may  then  be  obtained  readily  by  trigonometry. 
It  is  an  admirable  way  to  keep  records  and  working  maps. 
Steel  straight-edges  should  be  used  in  drawing  the  lines. 
To  project  a  line  too  long  to  be  drawn  with  the  straight- 
edge, drive  a  fine  needle  at  one  end  to  which  fasten  a  fine 
silk  thread.  Stretch  the  thread  exactly  over  the  line  and 
at  points  which  may  be  included  within  the  length  of  the 
straight-edge  make  fine  marks  on  the  paper.  Draw  the 
extended  line  through  these  marks. 

REPRODUCING   MAPS 

For  reproducing  drawings  smaller  than  12  ins.  by  18  ins. 
hektographs  or  clay  process  pads  are  good.  The  hekto- 
graph  is  a  pad  of  gelatine  in  a  shallow  pan,  the  clay  pad 


TRANSIT  SURVEYING  267 

being  a  substitute  for  gelatine  and  in  appearance  re- 
sembling putty.  The  drawings  are  made  with  specially 
prepared  hektograph  inks  on  a  smooth  hard  paper.  This 
is  laid  face  down  on  the  pad  and  rubbed  to  remove  wrinkles 
and  air  bubbles.  It  is  left  in  place  for  three  minutes. 
After  removal  plain  pieces  of  paper,  preferably  not  so  hard 
as  the  original,  are  laid  on  the  pad  in  rapid  succession  and 
rubbed  smooth.  Each  sheet  receives  a  print,  the  im- 
pressions becoming  gradually  more  faint.  Fifteen  to 
twenty  good  impressions  is  the  limit.  The  inks  can  be  had 
in  a  number  of  colors.  If  only  a  few  copies  are  wanted, 
less  than  six  or  seven,  the  original  drawing  may  be  made 
with  copying  pencils,  also  to  be  obtained  in  several  colors. 

For  larger  drawings  tracings  are  used.  Tracing  cloth  is 
costly,  and  medium  thick  "Parchment,"  "Vellum," 
"Colonna"  or  "Ionic"  tracing  papers  are  well  suited  to 
the  use  of  the  surveyor.  When  tracings  are  to  receive 
much  handling  or  are  expected  to  be  used  for  many  years 
they  should  be  on  tracing  cloth.  The  drawing  is  made  on 
paper  and  traced  in  ink  on  the  transparent  cloth  or  paper. 
To  save  time  on  unimportant  work  the  drawing  is  made 
directly  on  tracing  paper  or  on  the  rough  side  of  tracing 
cloth  in  lead  pencil  and  then  inked  in.  Tracing  cloth  and 
paper  are  greatly  affected  by  moisture  in  the  atmosphere 
so  it  is  best  to  make  accurate  maps  first  on  paper.  Sur- 
veyors should  use  a  good  quality  of  paper  for  maps. 

When  several  copies  of  a  drawing  are  wanted  the  tracing 
is  blue  printed.  Blue-print  papers  are  sold  by  all  dealers 
in  surveying  supplies.  In  a  frame  holding  a  sheet  of  plate 
glass,  the  tracing  is  placed  on  the  glass.  The  sensitized 
surface  of  a  sheet  of  blue-print  paper  is  laid  on  the  tracing. 
On  top  is  placed  a  piece  of  felt  or  blanket,  and  the  whole 
is  covered  with  a  wooden  cover  held  in  place  by  springs 
and  clamps.  All  bubbles  of  air  and  wrinkles  in  the  tracing, 
the  blue-print  paper  and  the  felt  must  be  eliminated.  The 
glass  is  exposed  to  the  direct  rays  of  the  sun  for  several 
minutes,  depending  on  the  sensitiveness  of  the  prepared 
paper.  The  blue-print  paper  is  afterwards  washed  in  clean 
water  until  white  lines  appear  on  a  blue  ground,  and  then 
hung  on  a  line  to  dry. 

Men  who  have  occasionally  to  make  blue  prints  find  it 


268  PRACTICAL   SURVEYING 

convenient  to  prepare  paper  for  their  own  use,  for  this 
material  does  not  have  good  keeping  properties.  Take 
one-half  ounce  each  of  potassium  ferrid-cyanide  and  cit- 
rate of  iron  and  ammonia;  dissolve  in  from  6  ozs.  to  8  ozs. 
of  clean  water.  Put  in  a  bottle  and  shake  thoroughly  un- 
til the  chemicals  are  dissolved,  which  process  requires  about 
10  minutes.  The  mixture  does  not  keep  well  so  a  fresh  lot 
should  be  prepared  for  each  job  of  printing.  Only  chemi- 
cally pure  (C.  P.)  materials  should  be  used  and  they  can 
be  bought  in  crystal  form  from  any  druggist.  Keep  in  a 
light-proof  bottle  with  ground-glass  stopper. 

Eight  ounces  of  the  mixture  will  coat  about  100  square 
feet  of  paper.  Lay  in  a  shallow  dish  a  piece  of  thin  cotton 
or  linen  cloth  and  pour  in  the  mixture.  Then  lift  up  the 
cloth  to  allow  the  liquid  to  strain  through  so  all  undis- 
solved  lumps  will  be  removed.  The  work  should  be  done 
in  a  room  lighted  by  a  red  or  orange  light.  Use  a  broad 
flat  brush  for  coating  the  paper,  which  should  be  a  good 
bond  paper  with  smooth  surface.  Hang  in  the  dark  until 
dry  and  keep  in  the  dark  until  used. 

White  lines  on  a  blue  ground  are  not  satisfactory  when 
a  map  is  to  be  colored,  but  with  special  inks  or  erasing  fluids 
white,  red  and  yellow  lines  may  be  drawn  on  blue  prints. 
Several  manufacturers  sell  paper  which  is  used  like  blue- 
print paper,  the  lines  coming  out  blue,  black  or  brown  on 
a  white  ground.  There  is  also  on  the  market  a  thin  paper 
which,  used  like  blue-print  paper,  shows  white  lines  on  a 
brown  ground.  Such  a  print  may  be  used  with  ordinary 
blue-print  paper,  as  a  photographic  negative,  the  result  be- 
ing a  print  with  blue  lines  on  a  white  ground. 

PRACTICAL  ASTRONOMY 

An  ephemeris  is  a  table  giving  the  place  of  a  planet  for 
a  number  of  successive  days.  The  solar  ephemeris  is  of 
value  to  surveyors  and  engineers,  for  by  observations  on 
the  sun  the  local  time  at  a  place  may  be  ascertained,  the 
latitude  found  and  the  true  meridian  determined. 

Practically  all  instrument  makers  issue  annually  small 
books  containing  the  solar  ephemeris  for  the  year,  together 
with  much  other  useful  information.  These  vest  pocket 


TRANSIT  SURVEYING  269 

books  are  sold  for  a  small  price,  usually  ten  cents.  From 
the  1912  edition  of  the  twenty-five  cent  book  issued  by 
Wm.  Ainsworth  &  Sons,  Denver,  Colo.,  the  following  con- 
cise description  of  practical  astronomical  work  has  been 
taken. 

The  Sun  is  the  center  of  the  solar  system,  remaining  con- 
stantly fixed  in  its  position,  although  often  spoken  of  as  in 
motion  around  the  earth. 

The  Earth  makes  a  complete  revolution  around  the  sun 
in  three  hundred  and  sixty-five  days,  five  hours,  forty-eight 
minutes  and  forty-six  seconds. 

It  also  rotates  about  an  imaginary  line  passing  through 
its  center,  and  termed  its  axis,  once  in  twenty-three  hours, 
fifty-six  minutes  and  four  seconds,  mean  time,  turning 
from  west  to  east. 

The  Poles  are  the  extremities  of  the  earth's  axis.  The  pole 
in  our  own  hemisphere,  known  as  the  North  Pole,  if  pro- 
duced indefinitely  toward  the  concave  surface  of  the  heavens, 
would  reach  the  North  Pole  of  the  heavens  a  point  situated 
near  the  Polar  star. 

The  Equator  is  an  imaginary  line  passing  around  the 
earth,  equidistant  from  the  poles,  and  in  a  plane  at  right 
angles  with  the  axis. 

If  the  plane  of  the  equator  be  produced  to  the  heavens, 
it  forms  what  is  termed  the  Celestial  Equator. 

The  Orbit  of  the  earth  is  the  path  in  which  it  moves  in 
making  its  yearly  revolution.  If  the  plane  of  this  orbit 
were  produced  to  the  heavens,  it  would  form  the  Ecliptic, 
or  the  sun's  apparent  path  in  the  heavens. 

The  earth's  axis  is  inclined  to  its  orbit  at  an  angle  of 
about  23°  27',  making  an  angle  of  the  same  amount  between 
the  earth's  orbit  and  its  equator,  or  between  the  Celestial 
Equator  and  the  Ecliptic. 

The  Equinoxes  are  the  two  points  in  which  the  Ecliptic 
and  the  Celestial  Equator  intersect  one  another. 

The  Horizon  of  a  place  is  the  surface  which  is  defined  by 
a  plane  supposed  to  pass  through  the  place  at  right  angles 
with  a  vertical  line,  and  to  bound  our  vision  at  the  surface 
of  the  earth.  The  horizon,  or  a  horizontal  surface,  is  de- 
termined by  the  surface  of  any  liquid  when  at  rest,  or  by 
the  spirit-levels  of  an  instrument. 


270  PRACTICAL  SURVEYING 

The  Zenith  of  any  place  is  the  point  directly  overhead,  in 
a  line  at  right  angles  with  the  horizon. 

The  Meridian  of  any  place  is  a  great  circle  passing  through 
the  zenith  of  a  place  and  the  poles  of  the  earth. 

The  Latitude  of  a  place  is  its  distance  north  or  south  of 
the  Equator,  measured  on  a  Meridian.  At  the  equator  the 
latitude  is  o°,  at  the  poles  90°. 

Refraction.  —  By  reason  of  the  atmosphere,  the  rays  of 
light  from  the  sun  are  bent  out  of  their  course,  so  as  to 
make  its  altitude  appear  greater  than  is  actually  the  case. 

The  amount  of  refraction  varies  according  to  the  altitude 
of  the  body  observed,  being  zero  when  it  is  in  the  zenith, 
about  one  minute  when  midway  from  the  zenith  to  the 
horizon,  and  almost  thirty-four  minutes  when  in  the 
horizon. 

The  Longitude  of  a  place  is  its  angular  distance  east  or 
west  of  a  given  place  taken  as  the  starting  point  or  first 
meridian;  it  is  measured  on  the  equator  or  on  any  parallel 
of  latitude  and  usually  from  Greenwich,  England. 

As  the  earth  makes  a  complete  rotation  upon  its  axis 
once  a  day,  every  point  on  its  surface  must  pass  over  360° 
in  twenty-four  hours,  or  15°  in  one  hour,  and  so  on  in  the 
same  ratio.  And  as  the  rotation  is  from  west  to  east,  the 
sun  would  come  to  the  meridian  of  every  place  15°  west 
of  Greenwich  just  one  hour  later  than  the  time  given  in 
the  Ephemeris  for  apparent  noon  at  that  place. 

To  an  observer  situated  at  Denver,  Colo.,  the  longitude 
of  which  is,  in  time,  seven  hours,  the  sun  would  come  to 
the  meridian  seven  hours  later  than  at  Greenwich,  and 
thus  when  it  was  12  M.  at  that  place  it  would  be  but  5  A.M. 
in  Denver. 

TIME 

A  Solar  Day  is  the  interval  of  time  between  two  suc- 
cessive upper  transits  of  the  sun  across  the  same  meridian. 
Solar  days  are  of  unequal  length.  A  mean  solar  day  is 
the  average  for  a  year. 

A  Sideral  Day  is  the  interval  of  time  between  two  suc- 
cessive upper  transits  of  a  fixed  star  across  the  same 
meridian;  it  is  invariable  and  is  equal  to  twenty- three 
hours,  fifty-six  minutes,  four  and  nine-hundredths  seconds 


TRANSIT  SURVEYING  271 

of  mean  solar  time.  The  earth  makes  one  complete 
rotation  on  its  axis  in  a  sideral  day. 

Mean  Noon.  —  A  clock  keeps  mean  solar  time  when  it 
divides  a  mean  solar  day  into  twenty-four  equal  parts  or 
hours.  Noon  as  shown  by  such  a  clock  is  mean  noon. 

Apparent  Noon  for  any  place  is  the  time  of  the  upper 
transit  of  the  sun  across  the  meridian  of  that  place;  it 
may  occur  several  minutes  earlier  or  later  than  mean  noon. 

Equation  of  Time. — The  column  headed  "Equation  of 
Time"  in  the  Ephemeris  shows  the  quantity  to  add  to 
or  subtract  from  mean  time  to  obtain  the  corresponding 
apparent  time. 

Standard  Time.  —  Since  November,  1883,  in  the  United 
States,  the  mean  solar  time  of  the  meridians  60,  75,  90, 
105,  and  1 20  west  of  Greenwich  is  standard  time.  The  time 
spaces  are  known  respectively  as  Colonial,  Eastern,  Central, 
Mountain  and  Pacific  time.  Each  differs  from  the  next  in 
time  by  one  hour.  Instead  of  employing  the  local  mean 
solar  time,  the  time  used  1s  the  mean  solar  time  at  the 
nearest  of  the  standard  meridians. 

Hour  Angle.  —  The  number  of  hours  from  the  meridian. 

To  Set  a  Watch  to  Mean  Local  Time  by  the  Sun.  —  Set  up 
the  transit  and  adjust  the  telescope  to  the  true  meridian, 
then  note  the  exact  time  that  the  center  of  the  sun's  image 
crosses  the  vertical  cross-hair.  This  is  apparent  noon,  and 
at  this  instant,  set -the  minute  hand  to  as  many  minutes 
before  12  as  the  equation  of  time  for  the  given  day  shows 
is  to  be  added  to  mean  time,  or  to  as  many  minutes  after 
12  as  it  shows  is  to  be  subtracted.  The  correction  may  be 
noted  in  case  it  is  not  desired  to  set  the  watch. 

DECLINATION 

The  Declination  of  the  sun  is  its  angular  distance  north 
or  south  of  the  celestial  equator;  when  the  sun  is  at  the 
equinoxes,  that  is,  about  the  2ist  of  March  and  the  2ist 
of  September  of  each  year,  its  declination  is  o,  or  it  is  said 
to  be  on  the  equator;  from  these  points  its  declination  in- 
creases from  day  to  day  and  from  hour  to  hour,  until  on 
the  2ist  of  June  and  the  2ist  of  December  it  is  23°  27' 
distant  from  the  equator. 


272  PRACTICAL  SURVEYING 

It  is  the  declination  which  causes  the  sun  to  appear  so 
much  higher  in  summer  than  in  winter,  its  altitude  in  the 
heavens  being  about  46°  54'  more  on  the  2ist  of  June  than 
it  is  on  the  2ist  of  December. 

The  Ephemeris  gives  the  sun's  declination  for  mean  noon 
at  Greenwich  for  each  day  in  the  year.  The  declination  of 
the  sun  at  any  place  for  any  hour  of  the  day  is  determined 
from  the  Ephemeris  as  follows: 

1.  Divide   the   longitude  of   the  place   (reckoned   from 
Greenwich)  by  15  to  obtain  the  corresponding  difference  of 
time  in  hours. 

2.  Find  the  corresponding  Greenwich   time  by  adding 
the  difference  of  time  to  the  mean  time  at  the  given  place, 
when  west  from  Greenwich,  and  subtracting  when  east. 

3.  Multiply  the  difference  for  one  hour,  as  found  in  the 
table  opposite  the  given  day  of  the  year,  by  the  number  of 
hours  from  noon  by  Greenwich  time. 

4.  This  product  is  the  change  in  declination  to  be  ap- 
plied as  indicated  in  the  following  expression: 


Declination  for 
given  time 
and  place 


{apparent  dec. 
for  given  day 
of  year 


change  in 
declina- 
tion. 


The  sign  of  the  last  term  is  +  for  time  after  noon,  Green- 
wich, when  declination  is  increasing,  and  for  time  before 
noon  when  declination  is  decreasing. 

The  —  sign  is  to  be  used  for  time  after  noon,  Greenwich, 
when  declination  is  decreasing  and  for  time  before  noon 
when  declination  is  increasing. 

An  inspection  of  the  Ephemeris  will  show  whether  the 
declination  is  increasing  or  decreasing  from  day  to  day. 

N  in  the  column  of  apparent  declination  indicates  north, 
and  S  indicates  south  declination. 

Example.  —  Required  the  declination  at  10  A.M.,  Aug. 
10,  1911,  at  Denver,  Colo.,  U.  S.  A.,  latitude  39°  46'  31" 
north,  longitude  105°  west. 

1.  Diff.  of  time  =  —   =  7  hrs. 

2.  As  Denver  is  west  from  Greenwich,  add  the  diff.  of 
time,  obtaining  10  A.M.,  Denver  time  =  10  -f-  7  =  17,  or  5 
P.M.,  Greenwich  time. 


TRANSIT  SURVEYING  273 

3.  Change  in  dec.  for  I  hr.  =  43". 32 
Change  for  5  hrs.             =  43". 32  X  5  =  o°  03'  36".6o 

4.  Sun's  apparent  dec.  at 

Greenwich,  mean  noon  =  N  15°  49'  53". 60 

Dec.  at  10  A.M.,  Denver  =  N  15°  46'  17" 

The  change  is  subtracted  as  the  time  is  afternoon  and 
dec.  is  decreasing,  since  the  dec.  the  next  day  is  less. 

LATITUDE 

Determination  of  Latitude  by  Direct  Observation  of  the  Sun. 
—  Carefully  level  the  transit  a  few  minutes  before  apparent 
noon,  and  if  it  is  not  provided  with  solar  hairs,  bring  the 
horizontal  cross-hair  tangent  to  the  upper  limb  of  the  sun 
and  keep  it  tangent  by  the  slow  motion  screw  until  the 
sun  ceases  to  rise,  then  read  the  vertical  angle,  and  from 
this  angle  subtract  the  semi-diameter  of  the  sun,  as  given 
in  the  Ephemeris  for  the  proper  month  of  the  year,  also 
the  refraction  corresponding  to  the  observed  angle.  The 
resulting  angle  will  be  the  true  altitude  of  the  sun's  center. 
Calculate  the  sun's  declination  for  noon,  apparent  time,  for 
place  of  observation  as  described  above.  If  the  declina- 
tion is  N,  subtract,  but  if  it  is  S,  add  it  to  the  true  altitude 
of  the  sun.  The  result  is  the  co-latitude  of  the  place  of 
observation  and  the  lat.  =  90°  —  co.-lat. 

Note.  —  In  direct  solar  observations  it  is  necessary  to 
protect  the  eye  by  a  darkened  glass,  which  should  be  used 
at  the  eye  end  of  the  telescope  unless  both  its  surfaces  are 
true  and  parallel.  When  the  altitude  is  high,  a  diagonal 
eyepiece  will  be  found  convenient,  or  an  image  of  the  sun 
and  cross-hairs  may  be  formed  on  a  screen,  a  card  or  a 
blank  page  of  a  notebook  held  a  few  inches  from  the  eye- 
piece. If  this  method  is  used  no  darkener  is  required.  An 
average  of  several  observations  is  preferable  to  a  single 
observation.  When  more  than  one  observation  is  made, 
the  alternate  ones  should  be  taken  with  the  telescope  re- 
versed in  order  to  eliminate  instrumental  errors.  When 
great  accuracy  is  not  essential,  standard  time  may  be  used 
in  computing  the  declination.  As  the  difference  between 
standard  and  local  time  is  seldom  more  than  30  min.  and 
the  greatest  hourly  change  in  declination  is  about  one 


274 


PRACTICAL  SURVEYING 


minute  of  an  arc,  the  maximum  error  in  declination  due  to 
using  standard  time  would  not  be  greater  than  30  seconds. 
Example.  —  On  April  17,  1910,  an  observation  was  made 
of  the  sun  to  determine  the  latitude  of  Denver.  The  hori- 
zontal cross-hair  was  kept  tangent  to  the  lower  limb  of  the 
sun  until  it  ceased  to  rise,  then  the  vertical  circle  reading 


Apparent  alt.  lower  limb 
Refraction  for  60°  20'  oo' 
True  alt.  lower  limb 
Add  sun's  semi-diameter 
True  alt.  sun's  center 
Subtract  declination 

Co-lat. 

Latitude 


=  60°  20'  oo" 
=  00°  oo'  34" 
=  60° 19' 26" 
=  00°  15'  58" 

=  60°  35'  24" 
=  10°  21'  55" 
=  50°  13'  29" 
=  39°  46' 3i" 


Latitude  may  be  determined  by  reference  to  an  accurate 
map,  from  which  the  latitude  of  a  neighboring  point  may 
be  found  and  then  due  allowance  made  for  the  distance 
north  or  south  to  the  station. 

The  table  on  page  275  gives  the  length,  in  feet,  of  one 
minute  of  arc  and  the  number  of  minutes  of  arc  in  a  mile 
for  latitude  and  longitude,  from  o°  to  60°  latitude  by  one- 
degree  intervals. 

MEAN  REFRACTION 

(To  be  Subtracted  from  Observed  Altitude,  in  Direct  Solar  Obser- 
vations.)    Barometer  30  Inches;  Thermometer  50°  F. 


Altitude. 

Refraction. 

Altitude. 

Refraction. 

10° 

5'  i9' 

20° 

2     39' 

11° 

4'  5i' 

25° 

2     04' 

12° 

4'  27' 

30° 

I     41' 

13° 

4'  07' 

35° 

I      23' 

14° 

3'  49' 

40° 

I     09' 

15° 

3'  34' 

45° 

58' 

16° 
17° 

3'  20' 
3'  08' 

£ 

49; 

34 

18° 

2'  57' 

70° 

ax' 

19° 

2'  48' 

80° 

tot 

TRANSIT  SURVEYING 


275 


TABLE  SHOWING  FEET  PER  MINUTE  (ARC)  AND  MINUTES  (ARC) 
PER  MILE  OF  LATITUDE  AND  LONGITUDE 


Length  of  I  1 

Vtin.  in  Feet. 

No.  of  Min 

.  in  i  Mile. 

Latitude. 

Latitude. 

Longitude. 

Latitude. 

Longitude. 

0° 

6045 

6087 

0-8734 

0.8674 

i° 

6045 

6085 

0-8734 

0.8677 

2° 

6045 

6083 

0.8734 

0.8679 

3° 

6045 

6078 

0.8734 

0.8687 

4° 

6045 

6071 

0-8734 

0.8697 

5° 

6045 

6063 

0.8734 

o  .  8708 

6° 

6045 

6053 

0.8734 

0.8722 

7° 

6046 

6041 

0-8733 

0.8740 

8° 

6046 

6027 

0.8733 

0.8760 

9° 

6046 

6oi2 

0-8733 

0.8782 

10° 

6047 

5994 

0.8731 

0.8808 

11° 

6047 

5975 

0.8731 

0.8836 

12° 

6048 

5954 

0.8730 

0.8867 

13° 

6048 

593i 

0.8730 

o  .  8902 

14° 

6049 

5907 

0.8728 

0.8938 

15° 

6049 

5880 

0.8728 

0.8979 

16° 

6050 

5852 

0.8727 

0.9022 

17° 

6050 

5822 

0.8727 

o  .  9069 

18° 

6051 

5790 

0.8725 

0.9119 

19° 

6052 

5757 

0.8724 

0.9171 

20° 

6052 

572i 

0.8724 

0.9229 

21° 

6053 

5684 

0.8722 

0.9280 

22° 

6054 

5646 

0.8721 

0.9351 

23° 

6054 

5605 

0.8721 

0.9420 

24° 

6055 

5563 

0.8720 

0.9491 

25° 

6056 

55i9 

0.8718 

0.9566 

26° 

6057 

5474 

0.8717 

0.9645 

27° 

6058 

5427 

0.8715 

0.9729  . 

28° 

6059 

5378 

0.8714 

0.9817 

29° 

6060 

5327 

0.8712 

0.9911 

30° 

6061 

5275 

0.8711 

.0009 

3i° 

6061 

5222 

0.8711 

.OIII 

32° 

6062 

5i66 

0.8709 

.0220 

33° 

6063 

5109 

0.8708 

•0334 

34° 

6064 

5051 

0.8707 

•0453 

35° 

6065 

499i 

0.8705 

•0579 

36° 

6066 

4930 

0.8704 

.0709 

37° 

6067 

4867 

0.8702 

.0848 

38° 

6068 

4802 

0.8701 

•  0995 

39° 

6070 

4736 

o  .  8698 

.1148 

40° 

6071 

4669 

0.8697 

.  1308 

4i° 

6072 

4600 

0.8695 

.1478 

42° 

6073 

4530 

0.8694 

.1655 

43° 

6074 

4458 

0.8692 

.1846 

276 


PRACTICAL  SURVEYING 


TABLE  SHOWING  FEET  PER  MINUTE  (ARC)  AND  MINUTES  (ARC) 
PER  MILE  OF  LATITUDE  AND  LONGITUDE  (Continued) 


Latitude. 

Length  of  i  Min.  in  Feet. 

No.  of  Min.  in  i  Mile. 

Latitude. 

Longitude. 

Latitude. 

Longitude. 

44° 

6075 

4385 

0.8691 

.  2040 

45° 

6076 

4311 

0.8689 

.2247 

46° 

6077 

4235 

0.8688 

.2467 

47° 

6078 

4158 

0.8687 

.2698 

48° 

6079 

4080 

0.8685 

.2941 

49° 

6080 

4OOI 

o  .  8684 

.3196 

50° 

6081 

3920 

0.8682 

•3469 

6082 

3838 

0.8681 

•3757 

52° 

6084 

3755 

0.8678 

.4061 

53° 

6085 

3671 

0.8677 

.4383 

54° 

6086 

3586 

0.8675 

•4723 

55° 

6087 

3499 

0.8674 

.5090 

56° 

6088 

3413 

0.8672 

•5470 

57° 

6089 

3323 

0.8671 

.5888 

58° 

6090 

3233 

0.8669 

•6331 

6091 

3142 

0.8668 

.6804 

6°° 

6092 

3051 

0.8667 

•7305 

ERRORS  IN  AZIMUTH  FOR  i  MINUTE  ERROR  IN  DECLINATION  OR 
LATITUDE 


For  i  Min.  Error  in  Declination. 

For  i  Min.  Error  in  Latitude. 

Lat.  30°. 

Lat.  40°. 

Lat.  50°. 

Lat.  30°. 

Lat.  40°. 

Lat.  50°. 

H  M 

Min. 

Min. 

Min. 

Min. 

Min. 

Min. 

o  30 

I    OO 

8.85 
4.46 

IO.OO 

5.05 

12.90 
6.01 

8-77 
4-33 

9-92 
4.87 

II.80 
5-80 

2   00 

2.31 

2.61 

3-H 

2.00 

2.26 

2.70 

3  oo 

I.63 

1.85 

2.  2O 

I-I5 

1.30 

1.56 

4  oo 

i-34 

1-51 

I.  80 

0.67 

0-75 

0.90 

5  oo 

i  .20 

1.35 

1.61 

0.31 

0-35 

0-37 

6  oo 

i.  IS 

1.30 

1.56 

o.oo 

0.00 

0.00 

By  the  use  of  the  above  table  the  amount  of  the  azimuth 
error,  resulting  from  the  use  of  erroneous  declination  or 
latitude  at  the  different  hours  of  the  day,  may  be  de- 
termined. 


TRANSIT  SURVEYING  277 

If  the  South  Polar  Distance  used  be  too  great,  the  ob- 
served meridian  falls  to  the  right  of  the  true  south  point  in 
the  forenoon  and  to  the  left  in  the  afternoon  and  vice  versa 
if  too  small. 

If  the  latitude  used  be  too  great,  the  observed  meridian 
falls  to  the  left  of  the  true  south  point  in  the  forenoon  and  to 
the  right  in  the  afternoon  and  vice  versa  if  too  small. 


DETERMINATION  OF  THE   MERIDIAN 

By  Direct  Solar  Observation.  —  The  best  time  of  day  for  a 
solar  observation  is  from  8  to  10  A.M.,  and  from  2  to  4  P.M. 
Observations  greater  than  five  or  less  than  one  hour  from 
noon  should  not  be  relied  upon.  Use  colored  glass  over 
eye  end. 

The  transit  should  be  accurately  adjusted  and  carefully 
leveled.  Set  the  limb  at  zero  when  the  telescope  is  di- 
rected to  some  convenient  mark,  then  with  the  lower 
motion  clamped,  point  the  telescope  to  the  sun  and  bring 
the  vertical  and  horizontal  cross-hairs  tangent  to  its  image, 
say  in  the  lower  left-hand  quarter;  read  the  vertical  circle 
and  the  limb.  Note  the  time,  quickly  reverse  the  tele- 
scope in  altitude  and  azimuth  and  again  bring  the"  cross- 
hairs tangent  to  the  image,  but  in  the  opposite  quarter; 
read  the  vertical  circle  and  the  limb.*  The  mean  of  the 
vertical  circle  readings  will  give  the  apparent  altitude  of 
the  center  of  the  sun.  The  mean  of  the  readings  on  the 
limb  will  give  the  angle  between  the  sun  and  the  selected 
mark. 

Formula  A  .  —  Let  Z  =  the  angle  between  the  sun  and 
the  meridian,  then  its  value  may  be  obtained  from  the 
equation 

cos  * 


sin  co-alt.  X  sin  co-lat. 
where  5  =  co-dec.  +  co-alt.  +  co-lat. 

*  When  the  transit  has  a  vertical  arc  instead  of  a  full  circle  the  in- 
tersection of  the  cross-hairs  may  be  directed  to  the  center  of  the  sun's 
image.  In  fact  this  method  is  common  even  with  full  circles  for  the 
error  thus  introduced,  if  any,  is  small.  —  MCC. 


278 


PRACTICAL  SURVEYING 


Example.  —  The  following  notes  are  of  an  observation 
made  at  Denver  at  10  A.M.,  April  17,  1910.  Latitude 
39°  46' 31" 


Telescope. 

Horizontal  angle. 

Vertical  angle. 

Position  of 
sun's  image. 

Direct  

89°  17'  R 

+  50°   14' 

,J. 

Reversed  

89°  07'  R 

+  51°  07' 

-P 

Reduction  of  notes.  — 10  A.M.,  Denver  time 
17  or  5  P.M.,  Greenwich  time. 
Change  in  dec.  since  noon 


10 


Greenwich 


Dec.  at  noon  Greenwich 
Dec.  at  10  A.M.  Denver 
co-dec. 


5  X  53    -02 


=  +00°  04' 25".  i 
=  N  10°  1 5'  44"  4 
=  N  10°  20'  09 
=       79°  39'  50 


50°  40'  30" 
00°  oo'  49" 

50°  39' 4i" 
39°  20'  19" 
39°  46' 3i" 
50° 13' 29" 

=  \  [(79°  39'  5<>".5)  +  (39°  20'  19")  +  (50°  13'  29")] 

84°36'49".2 


Subtract  refraction  correction        = 
True-alt. 
Co-alt. 
Lat. 
Co-lat.  = 


Substituting  in  equation  for  cos  J  Z 

cos  i  z  =  t/(sin  84°  36'  49".2)  X  (sin 
V      si 


58"  .7) 


(sin  39°  20'  19")  X  (sin  50°  13'  29") 

+9.9980780" 

+  8.9359106 

—  9.8020222 
.-9.8856776J 
65°io'.t8",  Z  =  130°  20'  36". 


=  9-6231444 


Interpretation  of  result. — Fig.  208  shows  the  relative  posi- 
tions of  the  sun,  the  meridian,  and  the  point  of  reference. 


TRANSIT  SURVEYING 


279 


Before  noon  Z  is  to  the  left  of  the  line  to  the  sun  and  to 
the  right  after  noon. 

The  true  bearing  of  the  reference  point  is  N  41°  08'  36"  E. 

Formula  B.  —  Identical  results  will  be  obtained  by  the 
use  of  the  following  equation  and  it  may  be  preferred  on 
account  of  its  containing  the  direct  angles  instead  of  the 
co-angles,  but  it  is  necessary  to  pay  careful  attention  to 
the  algebraic  signs  when  it  is  used, 


cos  O  = 


cos  lat.  X  cos  alt. 


—  tan  lat.  X  tan  alt. 


The  sign  of  the  first  term  of  the  right-hand  side  of  the 
equation  is  negative  when  declination  is  S ;  the  second  term 
is  positive  where  the  latitude  is  S.  If  the  algebraic  sign  of 


the  result  is  positive,  Q  is  the  angle  between  the  sun  and 
the  north  point,  but  if  it  is  negative,  it  is  the  angle  between 
the  sun  and  the  south  point. 

Example.  —  The  solution  of  the  above  example  by  this 
equation  is  as  follows: 


280  PRACTICAL  SURVEYING 


COS  Q  = 


sin  10°  20'  09"  .5 


(cos  39°  46'  31")  X  (cos  50°  39'  41 


-  (tan  39°  46'  31")  X  (tan  50°  39'  41 

log  sin  10°  20'  09"  .5  =  +  9.2538706 
log  cos  39°  46'  31"  =  +  9.8856775 
log  cos  50°  39'  41 "  =  —  9.8020222 

9.5661709  =  log  0.368274 
log  tan  39°  46'  31"      =  +  9.9203518 
log  tan  50°  39'  41 "      =  +  0.0863893 

0.0067411  =  log  1.015640 
cos  Q  =  -  0.647366 
•'•  Q  =  49°  39'  24" 

Interpretation  of  result.  —  Q  is  the  angle  between  the  sun 
and  the  south  point,  since  the  declination  is  N,  the  lati- 
tude is  N  and  the  algebraic  sign  of  the  right-hand  side  of 
the  equation  is  negative.  The  true  bearing  of  the  reference 
point  is  therefore  N  41°  08'  36"  E,  the  same  as  obtained  by 
the  first  equation;  see  Fig.  208. 

Time  may  also  be  determined  from  the  foregoing  data 
by  the  equation  (for  a  spherical  triangle) 

.     „      sin  Z  sin  co-alt. 

sin  1  = : —  —  > 

sin  co-dec. 

in  which  T  is  the  hour  angle  of  the  sun  at  the  time  of  ob- 
servation. This  is  reduced  to  time  by  dividing  by  15  and 
corrected  by  the  equation  of  time  to  obtain  mean  local 
time  and  still  further  corrected  by  the  difference  between 
mean  local  time  and  standard  time  if  the  latter  is  desired. 

Example.  —  Using  the  data  of  the  example  already  given 
to  find  the  time. 

.     T  _  (sin  130°  20'  36")  (sin  39°  20'  19") 

Sill    JL     —  /    •  f\          •  ff* 

(sin  79°  39'  50.5") 

Log  sin  130°  20'  36"  =  +9.8820570 
Log  sin    39°  20'  19"  =  +9.8020222 

9.6840792 

Log  sin  79°  39'  50.5"  =  -9.9928945 
Log  sin  T  =  9.6911847 

T  =  29°  24'  50" 


TRANSIT  SURVEYING  281 

.*.  Time  before  noon  =  -^—  -  =  I  h.  57  m.  39  s. 

Noon 12  h.  oo  m.  oo  s. 

Deduction I  h.  57  m.  39  s. 

Sun  time    10  h.  02  m.  21  s.  A.M. 

Equation  of  time  to  nearest  sec. ,  —  1 6  s. 

Mean  local  time 10  h.  02  m.  05  s.  A.M. 

As  mean  local  time  and  standard  time  are  the  same  at 
Denver  no  further  correction  is  necessary. 

According  to  the  above  result,  the  time  used  was  02  m. 
05  s.  slow,  making  an  error  of  about  02"  in  the  declination 
which,  from  the  table  on  page  276,  makes  an  error  in  the 
azimuth  of  about  05",  being  well  within  the  allowable 
limits  of  error  for  ordinary  field  work. 

Note.  —  The  signs  preceding  the  logarithms  in  the  three 
examples  above  are  used  to  indicate  whether  the  logarithm 
is  to  be  added  or  subtracted. 

THE  MERTOIOGRAPH 

Mr.  Louis  Ross,  a  civil  engineer  in  San  Francisco,  Cali- 
fornia, placed  on  the  market,  in  1913,  a  device  by  means  of 
which  the  true  north  may  be  found  at  any  time  of  the  day, 
without  any  computations  or  preparation  of  tables,  the 
only  instrumental  operation  being  an  observation  on  the 
sun  with  the  transit.  Between  10  A.M.  and  2  P.M.  the  re- 
sults are  unreliable,  for  the  best  time  is  when  the  altitude 
of  the  sun  is  from  15°  to  25°.  Results  obtained  when  the 
altitude  is  between  25°  and  35°  are  only  fair,  while  an  alti- 
tude of  more  than  45°  should  not  be  used.  The  fore- 
going remarks  apply  as  well  to  direct  observations  on  the 
sun  when  the  solution  of  a  spherical  triangle  is  made  as 
previously  described. 

The  surveyor  needs  a  table  of  the  sun's  ephemeris,  but 
instead  of  going  to  the  labor  of  making  a  table  for  the 
declination  for  each  hour  he  merely  takes  from  the  table 
the  declination  for  Greenwich  time.  A  diagram  ac- 
companying the  meridiograph  is  then  consulted  and  the 
declination  for  the  time  and  place  is  at  once  obtained. 
Another  diagram  gives  the  refraction  to  be  subtracted. 
Declination  is  D. 


282  PRACTICAL   SURVEYING 

The  latitude  of  the  place  may  be  taken  from  a  good  map. 
If  not  available  measure  at  noon  the  sun's  altitude  H;  then, 
if  declination  is  north,  Latitude  =  90°  —  H  -f-  D;  if  declina- 
tion is  south,  Latitude  =  90°  —  H  —  D. 

"Apparent  local  time"  can  be  found  with  the  meridio- 
graph,  without  any  added  work,  by  a  single  setting  of  the 
proper  scales  nearly  as  fast  as  by  referring  to  a  watch  and 
to  an  accuracy,  so  the  inventor  claims,  of  one-half  minute 
or  less. 

From  a  circular  issued  by  Mr.  Ross,  the  following  de- 
scription is  taken,  which  should  be  clear  to  the  reader  by 
referring  to  the  illustration. 

The  meridiograph  is  a  protractor  which  solves  graphi- 
cally the  problem  of  finding  a  true  meridian  by  the  sun. 
It  consists  of  two  circular  scaled  disks,  7  in.  max.  diam., 
with  a  reading  arm.  The  names  of  all  scales  are  on  the 
arm,  exactly  over  the  graduations;  --  this  obviates  search- 
ing for  any  desired  scales.  Nearly  all  graduations  are  5 
or  10  minutes  spaces;  angles  may  therefore  be  read  to  an 
accuracy  of  I  minute. 

The  scales,  beginning  with  the  outermost,  are: 

NUMBER  o.io  to  i.oo,  complete  circle,  for  numbers  A,  B,  (A  +  B). 
BEARING  20°  to  88°  40'  approx.  if  loops,  for  true  bearing  of  sun. 

b      10°  to  60°         approx.    \  loop,  for  either  alt.  or  lat. 

a      10°  to  60°         approx.  I    loop,  for  either  alt.  or  lat. 

a      10°  to  60°          approx.  I     loop,  for  either  alt.  or  lat. 

b      10°  to  60°         approx.    \  loop,  for  either  alt.  or  lat. 
DECL.         i°  to  23°  30'  approx.  \\  loops,  for  declination. 

The  transparent  celluloid  cover  prevents  the  disks  from 
moving  accidentally  after  they  have  been  set  and  also  pro- 
tects the  scales  from  wear  and  dirt.  The  inner  disk  is  ro- 
tated through  the  finger  slots  above  and  below. 

To  find  true  north,  measure  sun's  altitude,  take  its 
declination  from  the  Ephemeris,  and  take  latitude  of 
station  from  a  map.  Set  these  data  on  the  meridiograph 
by  means  of  the  reading  arm,  thus: 

On  scales  a  set  alt.  against  lat.,  opposite  index  read  number  A. 

On  scales  b  set  alt.  against  lat.,  opposite  given  declination  read  number  B. 

Opposite  (A  -+-  B)  read  true  bearing  of  sun. 

On  the  two  a  scales  the  alt.  is  set  against  the  lat. ;    it  is 


TRANSIT  SURVEYING  283 

immaterial  which  is  set  on  which  as  the  scales  are  identi- 
cal, but  set  arm  on  outer  disk  first,  then  turn  inner  disk 
until  proper  reading  is  under  reading  line.  The  same  applies 
to  the  two  b  scales.  Note  that  the  two  a  scales  produce 
number  A,  while  the  Decl.  with  the  two  b  scales  produce 
number  B. 

Check  solution  of  example  given  on  face  of   meridio- 
graph. 


FIG.  209.    The  Meridiograph. 

The  solar  compass  was  superseded  by  the  solar  attach- 
ment to  the  transit  but  the  method  of  direct  observation  on 
the  sun  is  preferred  by  the  majority  of  surveyors.  The 
computations  however  are  tedious  and  the  amount  of 
figuring  required  makes  many  men  slight  the  work  some- 
what. With  the  most  careful  work  the  true  north  is  found 


284  PRACTICAL  SURVEYING 

only  within  the  nearest  I  or  2  minutes  of  angle.  The 
meridiograph  is  claimed  to  give  results  within  this  degree 
of  accuracy. 

BY  OBSERVATIONS   OF  POLARIS 

At  Elongation. — A  few  minutes  before  elongation,  set 
up  the  transit  and  center  the  plumb-bob  over  a  tack  driven 
into  a  stake.  Level  up  very  carefully  and  keep  the  verti- 
cal cross-hair  on  Polaris,  using  the  tangent  screw  of  the 
vernier  plate,  until  elongation  is  reached.  This  is  easily 
recognized  since  the  azimuth  then  remains  practically  con- 
stant for  several  minutes. 

When  elongation  has  been  reached,  depress  the  telescope 
and  carefully  fix  a  stake  on  line,  reverse  the  telescope  on 
its  axis  and  rotate  the  instrument  180°  on  its  vertical  axis, 
fix  the  vertical  hair  on  Polaris,  depress  the  telescope  and 
fix  another  stake  on  line,  if  the  vertical  hair  does  not  bisect 
the  first  one.  These  two  observations  must  be  made  be- 
fore Polaris  has  appreciably  commenced  its  return  motion 
in  azimuth. 

When  it  is  necessary  to  set  two  stakes,*  a  third  stake 
midway  between  them  will  be  in  a  vertical  plane  through 
the  plumb  line  of  the  transit  and  Polaris  at  elongation. 
By  daylight,  lay  off  from  this  plane  the  proper  azimuth. 
North  is  to  the  right,  if  Polaris  was  at  western,  and  to  the 
left,  if  at  eastern,  elongation. 

In  making  this  and  the  following  observation  it  is  neces- 
sary to  illuminate  the  stakes  and  the  cross-hairs.  The 
latter  may  be  accomplished  by  a  suitable  lamp  held  at  one 
side  of  the  transit,  so  that  sufficient  light  is  reflected  into 
the  telescope. 

At  Culmination.  —  On  account  of  the  great  difficulties  at- 
tending this  method,  it  should  be  used  only  when  the 
method  of  elongation  is  impracticable. 

This  method  is  based  on  the  fact  that  Polaris  is  very 
nearly  on  the  meridian  when  it  is  in  the  same  vertical  plane 
with  the  star  Delta,  in  the  constellation  Cassiopeia,  or 

*  When  the  transit  is  in  correct  adjustment  it  will  be  necessary  to 
set  but  one  stake,  whose  position  will  correspond  with  that  of  the  third 
one,  as  given  above. 


TRANSIT   SURVEYING 


285 


mOt '  ^80-LOQ  inoav 


JLV3i/f) 


•JO 


POLARIS. 

X] 


Zeta  of  the  Great  Dipper,  the  star  at  the  bend  in  the  handle. 
It  consists  in  watching  either  Delta  or  Zeta  until  it  comes 
into  the  same  vertical  plane  with  Polaris  and  then  waiting 
a  known  interval  of  time,*  until  Polaris  is  on  the  meridian. 
The  vertical  cross-hair  must  be  placed  on  Polaris  precisely 
at  the  end  of  this  interval  as  the  motion,  in  azimuth,  is 
most  rapid  at  culmination.  The  telescope  is  now  in  the 
meridian,  which  may  be  marked  in  any  suitable  manner. 

Limitations.  —  On  account  of  the 
haziness  of  the  atmosphere  near 
the  horizon,  the  lower  culminations 
of  Zeta  and  Delta  cannot  be  used 
below  about  38°  north  latitude; 
neither  can  their  upper  culmina- 
tions  be  used  north  of  about  25° 
and  30°  respectively,  on  account  of 
their  being  too  near  the  zenith. 

Selecting  the  Star.  —  The  dia- 
gram,  Fig.  209,  shows  Delta  Cassi- 
opeia  on  the  meridian  below  the 
pole  at  midnight  about  April  10. 
It  may  therefore  be  used  in  the 
above  method  for  two  to  three 
months  before  and  after  that  date. 
Likewise  Zeta,  of  the  Great  Dipper, 
may  be  used  for  two  to  three 
months  before  and  after  October  10. 

Time  of  Elongation  and  Culmina- 
tion.—  Fig.  210  shows  Polaris  near 
eastern  elongation  at  midnight  about  July  10,  at  western 
at  midnight  about  January  10,  at  upper  culmination  at 
midnight  about  October  10,  and  at  lower  at  midnight  about 
April  10.  The  approximate  time  of  elongation  or  culmi- 
nation for  other  dates  may  be  determined  by  noting  the 
position  of  the  line  adjoining  Zeta  of  the  Great  Dipper 
and  Delta  Cassiopeia.  When  this  line  is  vertical,  Polaris 
is  near  culmination  and  when  horizontal  it  is  near  elonga- 
tion. Polaris  is  on  the  opposite  side  of  the  Pole  from  Zeta 
of  the  Great  Dipper,  thus  furnishing  a  convenient  means 

*  The  waiting  time  for  1912  is  6  min.  48  sec.  for  Zeta  of  the  Great 
Dipper  and  7  min.  21  sec.  for  Delta  Cassiopeia. 


Directior 


notion 


of  apparent 


Vr,' 

tr» 


IOPEIA 


C*SS 

MIDNIGHT  ABOUT  APRIL  JO  TH 
FIG.  210. 


286  PRACTICAL  SURVEYING 

of  distinguishing  eastern  from  western  elongation  and  upper 
from  lower  culmination.  When  Zeta  is  west,  Polaris  is  east 
of  the  pole,  and  when  Zeta  is  above,  Polaris  is  below  the 
pole. 

MEAN  POLAR  DISTANCE 

The  azimuth  of  Polaris  at  elongation  for  any  year  is 
found  from  the  following  table,  which  gives  the  mean 
polar  distance.  The  maximum  error  in  azimuth  for  any 
month  cannot  exceed  one  minute  from  the  mean  polar 
distance  for  the  year,  this  being  within  the  limits  of  per- 
missible error. 

The  sine  of  the  true  azimuth  in  any  latitude  is  found  by 
the  formula. 

~ .     _  Sine  of  Polar  Distance 
Cosine  Latitude 


1915 
1916 
1917, 
1918, 
1919, 
1920, 


09  53-51 

08'  34.97" 

08'  16.45" 

07'  57-94" 

07'  39-45" 

07  20.98 


In  Engineering  News,  April  22,  1915,  appeared  the  fol- 
lowing article : 

AZIMUTH  OBSERVATIONS  ON  POLARIS  BY  DAYLIGHT 

By  ROBERT  V.  R.  REYNOLDS* 

Polaris  may  always  be  found  in  clear  weather  as  soon 
as  the  sun  has  set,  and  very  frequently  for  five  or  ten  min- 
utes before  sunset  or  after  sunrise.  It  is  stated  on  good 
authority  that  under  very  favorable  conditions  an  observa- 
tion has  been  successful  as  late  as  10  A.M.  In  the  northern 
parts  of  the  United  States  the  cross-wires  may  remain 
visible  for  a  long  time  after  sunset. 

For  the  novice  it  is  often  difficult  at  the  first  few  attempts 
to  see  Polaris  while  the  sky  is  still  bright,  but  having  once 

*  Forest  Examiner,  U.  S.  Forest  Service,  Washington,  D.  C. 


TRANSIT  SURVEYING 


287 


found  the  star,  which  appears  as  a  small  white  dot  in  the 
field,  he  will  never  thereafter  feel  in  doubt.     Tabulated 


TABLE  TO  FIND  POLARIS  BY  DAYLIGHT 


Hour  Angle  of  Polaris 
Approximate*  (Use  Suit- 
able Interpolations) 

Azimuth  Setting 
N  E  or  N  W 
Depending  upon  Position 
of  Polaris  E  or  W  of 
Meridian 

Altitude  Setting 
Latitude  Plus  or  Minus 
the  Tabulated 
Quantities 

o  .  o  or  1  2  .  o 

oo' 

08' 

0.5  or  11.5 

12' 

-1         07'  i 

i  .  o  or  1  1  .  o 

24' 

9            06'    * 

1.5  or  10.5 

36' 

*        °*'  -5. 

2.0  or  10.0 

47' 

J3                       OO        13^ 

2.5  or    9.5 

57' 

o                    55'      Jjf 

3  .  o  or    9.0 

1°  06' 

«j                 5o'     .3  "8 

3-5  or    8.5 

°     J5' 

/      ,-  u 

4  .  o  or    8.0 

0     22' 

~    C                            35'         ^  OT 

4-5  or    7.5 

°     27' 

"o"2            27'    |-ii 

5  .  o  or    7.0 

°    3°' 

*"  i                   I^'       **  rt 

5-5  or    6.5 
6.0  hours 

°    33' 

°     35' 

3"         S'  ^ 

*  The  hour  angle  used  as  the  argument  in  this  table  needs  only  to  be  approximate.  If 
it  is  correct  within  5  min.  of  time,  sufficiently  accurate  settings  will  be  indicated  provided 
interpolation  is  made.  Hence,  there  is  no  need  of  correcting  for  longitude  until  the  sur- 
veyor has  made  the  observation  and  is  preparing  to  enter  the  table  of  Azimuths  of  Polaris. 

The  table  is  computed  for  a  mean  latitude  of  42°,  but  purposely  modified  slightly  to 
make  it  more  useful  along  the  49th  parallel,  where  much  of  the  Forest  Service  work  is 
being  done.  It  is  accurate  enough  to  bring  Polaris  within  the  field  of  an  ordinary  transit 
between  latitudes  of  10°  and  58°  N.  It  is  not  for  use  to  determine  the  true  azimuth  after 
the  observations. 

settings  (such  as  accompany  this  article)  sufficiently  accu- 
rate to  bring  the  star  into  the  field  are  required,  and  there 
remain  several  factors  which  must  be  given  consideration 
before  success  can  be  assured: 

1.  A  slight  haziness,  which  may  hardly  be  obvious  to  the 
eye,  is  sufficient  to  conceal  the  star  until  darkness  comes 
on. 

2.  The  telescope  must  be  in  exact  focus  for  celestial  ob- 
jects.    This  may  be  accomplished  either  by  focusing  at 
night  upon  the  moon  and  making  a  slight  scratch  upon  the 
objective  slide  to  show  the  point  to  which  it  should  be  ex- 
tended, or  the  surveyor  may  focus  at  the  time  of  observa- 


288  PRACTICAL  SURVEYING 

tion  upon  a  well-defined  object  3  or  4  miles  distant,  which 
focus  will  usually  be  found  sufficiently  close.  Accurate 
focusing  is  one  of  the  most  important  factors  in  finding 
the  star. 

3.  For  the  purpose  of  cutting  off  objectionable  light,  the 
sunshade  should  always  be  attached.     Certainty  of  finding 
the  star  is  assured  by  throwing  a  coat  or  other  dark  cloth 
over  the  head  when  searching  through  the  telescope,  as  a 
photographer  uses  a  focusing  cloth. 

4.  An  approximate  meridian  must  be  had,  from  which 
the  azimuth  settings  are  turned  off.     Commonly,  the  sur- 
veyor will  already  have  such  a  meridian  from  his  backsight. 
Otherwise,  a  meridian  determination  from  a  solar  attach- 
ment in  reasonable   adjustment  will   suffice.     Sometimes, 
when  the  magnetic  declination  is  closely  known,  it  wTill  even 
be  possible  to  turn  upon  the  star  from  the  needle.     A  refer- 
ence meridian  which  is  true  within  5'  or  10'  will  be  precise 
enough  to  locate  the  star  when  the  table  of  approximate 
settings  is  used. 

Polaris  having  been  found,  the  angle  from  the  reference 
mark  to  the  star  should  be  measured  twice,  the  second 
time  with  the  telescope  inverted.  The  mean  time  of  obser- 
vation and  the  mean  angle  are  then  used  to  find  the  azimuth 
of  the  mark  by  the  simplified  hour-angle  method.  There  is 
practically  no  chance  that  any  other  star  will  be  seen  and 
mistaken  for  Polaris.  (Finis.) 


BY  ANY   STAR  AT  EQUAL  ALTITUDES 

In  high  latitudes  neither  the  sun  nor  Polaris  give  re- 
liable results.  The  sun  is  low  and  the  refraction  is  un- 
certain, while,  on  account  of  the  height  of  Polaris,  the 
observation  is  difficult  to  make  and  instrumental  errors  are 
magnified.  In  southern  latitudes  Polaris  is  not  visible  at 
all.  Although  this  method  may  be  used  in  any  latitude,  it 
is  particularly  applicable  under  the  above  conditions. 

The  method  consists  in  observing  a  star,  when  at  equal 
altitudes,  east  and  west  of  the  meridian.  The  meridian 
will  then  be  halfway  between  these  two  positions  of  the 
star.  The  star  selected  should  be  30°  or  more  from  the 
zenith  when  on  the  meridian  and,  at  least,  the  same  dis- 


TRANSIT  SURVEYING  289 

tance  from  the  pole.  The  observations  should  be  made 
when  the  star  is  three  to  four  hours  from  the  meridian. 

Making  the  observation.  —  To  make  the  first  observation 
level  the  transit  carefully,  direct  the  telescope  to  the  star, 
clamp  all  motions  and  fix  the  intersection  of  the  cross-hairs 
on  the  star  by  the  slow  motion  screws,  read  the  star's 
altitude,  unclamp  the  telescope  axis,  depress  the  telescope 
and  fix  a  point  on  line. 

To  make  the  second  observation  re-level  the  transit  care- 
fully, set  the  telescope  at  the  altitude  determined  by  the 
first  observation,  clamp  the  limb  and  lower  motion  when 
the  star  comes  near  to  the  horizontal  hair,  keeping  the  ver- 
tical hair  on  the  star  by  means  of  the  slow-motion  screw  of 
the  vernier  plate  until  it  reaches  the  intersection  of  the 
cross-hairs,  then  unclamp  the  telescope  axis,  depress  the 
telescope  and  fix  a  point  on  line.  A  third  point  set  half- 
way between  these  two  will  mark  the  meridian  through  the 
transit. 

If  the  transit  has  a  vertical  circle,  the  error  due  to  the 
adjustment  of  the  height  of  standards  may  be  eliminated 
by  making  the  first  observation  with  the  telescope  direct 
and  the  second  with  it  reversed  in  altitude  and  azimuth. 
In  this  method  artificial  illumination  must  be  used  for  the 
cross-hairs  and  for  setting  the  points  on  line. 

GREEK  LETTERS 

of    Alpha  e  Epsilon 

ft    Beta  17  Eta 

5  A    Delta  f  Zeta 

7    Gamma  *c  Kappa 


CHAPTER    VII 

SURVEYING  LAW  AND  PRACTICE 

The  business  of  the  surveyor  is  firmly  bound  up  with  that 
of  the  lawyer,  but  much  of  the  litigation  over  land  lines 
would  be  eliminated  if  the  lawyer  could  be  prevented  from 
interfering  with  the  surveyor  in  the  doing  of  his  work. 
When  it  comes  to  the  courts  that  is  another  matter,  for 
the  decisions  of  judges  based  upon  precedent,  and  latterly 
upon  changes  in  customs  and  advances  in  civilization,  are 
generally  right.  Even  if  occasionally  wrong  a  decision 
must  stand  until  a  better  informed  judge  has  a  similar  case 
presented  to  him.  The  decisions  of  courts  are  based  upon 
a  very  few  common  sense  principles  of  law,  but  it  should 
not  be  necessary  to  have  cases  go  into  court  in  order  to 
settle  lines  and  the  location  of  monuments. 

The  surveyor  is  presumed  to  have  enough  skill  to  measure 
angles  and  lines  and  make  a  record  of  same  so  competent 
surveyors  can  re-trace  the  work  and  re-locate  boundaries. 
The  average  attorney-at-law  does  not  recognize  "THE 
ERROR,"  a  thing  that  looms  up  large  before  every  surveyor 
and  which  the  courts  must  recognize. 

Mr.  J.  Francis  Le  Baron,  Member  of  the  American  So- 
ciety of  Civil  Engineers,  on  page  425,  in  the  Transactions 
of  the  American  Society  of  Civil  Engineers,  Vol.  LXXV 
(1912),  presents  the  following  as  a  sample  of  instructions 
for  re-surveys,  given  him  by  lawyers,  instructions  that  are 
positively  insulting  to  men  of  extended  experience  in  such 
work.  They  resemble  similar  instructions  which  lawyers 
at  times  attempted  to  give  the  author  in  the  years  when 
his  work  was  closely  associated  with  land  surveying. 

"I  want  you  to  start  from  the  beginning  corner,  as  given 
in  the  deed,  run  the  exact  course  and  distance  and  set  a 
stake  there.  It  will  not  take  you  long.  I  suppose  you 
make  an  allowance  for  the  variation  of  the  needle.  The 

290 


SURVEYING  LAW  AND   PRACTICE  2  91 

needle,  you  know,  does  not  point  exactly  north,  and  you 
must  make  an  allowance.  This  deed  reads  'due  north' 
so  many  chains.  Now  does  that  mean  true  north  or  the 
way  the  needle  points  ?  I  suppose  when  you  chain  down 
hill  you  make  an  allowance,  don't  you,  because  I  think 
the  distance  wouldn't  come  right  if  you  didn't?  I  don't 
know  how  much  you  allow,  but  I  suppose  you  have  some 
custom  about  it.  You  see  this  distance  reads  so  many 
chains  and  links,  so  you  must  measure  it  with  a  chain  and 
links,  and  not  any  other  kind  of  measure,  or  I  am  inclined 
to  think  that  the  Court  would  reject  your  survey." 

The  surveyor  is  also  employed  by  real  estate  agents  to 
make  surveys  for  clients  and  fully  ninety-five  per  cent  of 
these  real  estate  agents  demand  a  commission  of  no  small 
amount.  This  is  even  customary  with  some  attorneys  and 
managers  of  estates.  The  surveyor  who  is  compelled  to 
obtain  work  under  such  conditions  cannot  afford  to  do  the 
work  properly  and  furthermore  few  competent  surveyors 
will  accept  work  on  such  terms.  The  foregoing  remarks 
do  not  indict  all  lawyers,  real  estate  agents  and  managers 
of  estates,  for  many  high-minded  men  occupy  positions  of 
trust  and  jealously  safeguard  the  interests  of  their  clients 
to  a  degree  that  they  would  not  in  work  for  themselves. 
To  work  for  an  intelligent  and  honest  man  occupying  such 
a  position  is  an  experience  so  filled  with  satisfaction  that 
the  surveyor  often  feels  a  trifle  shamefaced  in  presenting 
his  bill.  If  he  could  afford  it  he  would  do  the  work  for 
the  mere  satisfaction  it  affords,  yet  work  done  for  such 
men  is  generally  the  most  highly  paid.  They  are  high- 
priced  themselves  and  prefer  to  have  work  done  by  men  who 
set  a  proper  value  upon  their  services.  There  is  no  heart- 
breaking competition  in  such  work,  yet  it  is  seldom  offered 
to  a  man  until  after  he  has  served  an  apprenticeship  of 
many  years  in  competitive  work  and  has  established  a 
reputation  for  accuracy  and  honesty. 

To  properly  re-survey  a  piece  of  ground  requires  consider- 
able preliminary  work  in  obtaining  starting  points,  for  in  a 
community  every  lot  is  tied  to  'adjoining  lots.  Some- 
times to  re-locate  one  line,  means  the  running  of  many  lines, 
some  at  a  considerable  distance  from  the  one  wanted. 
Ma\iy  men  do  not  want  to  pay  for  such  work.  Once  the 


2Q2  PRACTICAL  SURVEYING 

author  had  to  work  for  ten  days  before  he  could  definitely 
locate  the  line  he  had  been  employed  to  survey,  this  work 
consuming  only  three  hours'  time.  When  his  bill  was  pre- 
sented the  lawyer  offered  him  pay  for  the  three  hours  and 
told  him  to  go  to  the  other  owners,  whose  land  he  had  first 
run  out,  for  the  remainder  of  the  bill.  It  took  a  lawsuit  to 
recover  the  amount  due  and  some  evidence  obtained  in 
the  suit  resulted  in  the  disbarment  of  the  attorney.  This 
leads  naturally  to  the  statement  that  not  every  attorney 
is  a  lawyer  and  with  real  lawyers  there  is  seldom  any  diffi- 
culty. The  average  attorney,  therefore,  is  the  man  who  tries 
to  hold  the  surveyor  down  to  a  definite  procedure  and  who 
is  ignorant  of  THE  ERROR.  It  is  difficult  however  to  dis- 
tinguish between  the  real  lawyer  and  the  pseudo-lawyer, 
who  is  really  only  a  licensed  attorney-at-law,  until  some 
experience  is  had  with  the  man. 

Every  student  who  contemplates  following  surveying  as 
a  vocation  should  procure  a  copy  of  Paper  No.  1242,  Trans- 
actions of  the  American  Society  of  Civil  Engineers,  entitled 
"Retracement  Surveys  —  Court  Decisions  and  Field  Pro- 
cedure," by  N.  N.  Sweitzer,  M.  Am.  Soc.  C.  E.,  together 
with  the  complete  discussion  by  a  number  of  experienced 
men.  He  should  also  read  "Boundaries  and  Landmarks," 
by  A.  C.  Mulford  ($1.00)  and  "Descriptions  of  Lands,"  by 
R.  W.  Cautley  ($1.00)  in  order  to  obtain  the  point  of  view 
of  competent  surveyors  and  hasten  the  acquirement  of 
knowledge  he  can  obtain  otherwise  only  by  years  of  expe- 
rience. The  surveyor  must  not  forget  that  his  work  is  not 
to  merely  re- trace  old  lines  but  to  FIND  THE  LAND.  To  re- 
run old  field  notes  exactly  as  recorded  is  a  simple  matter, 
but  to  determine  the  location  of  the  original  monuments 
and  lines  calls  for  skill  gained  only  by  practical  experience. 
The  surveyor's  work  is  as  much  legal  as  technical  and  many 
•times  the  legal  side  is  the  most  important.  The  importance 
of  the  legal  nature  of  a  surveyor's  work  is  so  great  that  a 
number  of  years  ago  F.  Hodgman,  then  secretary  of  the 
Michigan  Engineering  Society,  wrote  a  book  giving  the  gist 
of  a  number  of  court  decisions  as  a  guide  for  surveyors. 
This  work  should  be  in  the  possession  of  every  land  sur- 
veyor. It  is  entitled  UA  Manual  of  Land  Surveying" 
($2.50).  John  Cassan  Wait  also  wrote  a  book  entitled 


SURVEYING  LAW  AND   PRACTICE  293 

"Law  of  Operations  Preliminary  to  Construction  in  En- 
gineering and  Architecture,"  which  covers  very  completely 
the  subject  of  the  law  of  boundaries. 

A  number  of  years  ago  Chief  Justice  Cooley,  of  the  Mich- 
igan Supreme  Court,  delivered  an  address  at  a  meeting  of 
the  Michigan  Engineering  Society  on  "The  Judicial  Func- 
tions of  Surveyors,"  and  no  modern  textbook  on  surveying 
is  presumed  to  be  complete  unless  this  address  is  made  a 
part  of  the  contents. 

THE   JUDICIAL  FUNCTIONS   OF   SURVEYORS 

By  CHIEF  JUSTICE  COOLEY 

When  a  man  has  had  a  training  in  one  of  the  exact 
sciences,  where  every  problem  within  its  purview  is  sup- 
posed to  be  susceptible  of  accurate  solution,  he  is  likely  to 
be  not  a  little  impatient  when  he  is  told  that,  under  some 
circumstances,  he  must  recognize  inaccuracies,  and  govern 
his  action  by  facts  which  lead  him  away  from  the  results 
which  theoretically  he  ought  to  reach.  Observation  war- 
rants us  in  saying  that  this  remark  may  frequently  be  made 
of  surveyors. 

In  the  State  of  Michigan  all  our  lands  are  supposed  to 
have  been  surveyed  once  or  more,  and  permanent  monu- 
ments fixed  to  determine  the  boundaries  of  those  who  should 
become  proprietors.  The  United  States,  as  original  owner, 
caused  them  all  to  be  surveyed  once  by  sworn  officers,  and 
as  the  plan  of  subdivision  was  simple,  and  was  uniform 
over  a  large  extent  of  territory,  there  should  have  been, 
with  due  care,  few  or  no  mistakes,  and  long  rows  of  monu- 
ments should  have  been  perfect  guides  to  the  place  of  any 
one  that  chanced  to  be  missing.  The  truth  unfortunately 
is,  that  the  lines  were  very  carelessly  run,  the  monuments 
inaccurately  placed,  and  as  the  recorded  witnesses  to  these 
were  many  times  wanting  in  permanency,  it  is  often  the 
case  that  when  a  monument  was  not  correctly  placed,  it  is 
impossible  to  determine  by  the  record,  by  the  aid  of  any- 
thing on  the  ground,  where  it  was  located.  The  incorrect 
record  of  course  becomes  worse  than  useless  when  the 
witnesses  it  refers  to  have  disappeared. 

It  is,  perhaps,  generally  supposed  that  our  town  plats 


294  PRACTICAL   SURVEYING 

were  more  accurately  surveyed,  as  indeed  they  should  have 
been,  for  in  general  there  can  have  been  no  difficulty  in 
making  them  sufficiently  perfect  for  all  practical  purposes. 
Many  of  them  however  were  laid  out  in  the  woods;  some 
of  them  by  proprietors  themselves,  without  either  chain  or 
compass,  and  some  by  imperfectly  trained  surveyors,  who, 
when  land  was  cheap,  did  not  appreciate  the  importance 
of  having  correct  lines  to  determine  boundaries  when  land 
should  become  dear.  The  fact  probably  is,  that  town  sur- 
veys are  quite  as  inaccurate  as  those  made  under  authority 
of  the  general  government. 

It  is  now  upwards  of  fifty  years  since  a  major  part  of  the 
public  surveys  in  what  is  now  the  State  of  Michigan  were 
made  under  the  authority  of  the  United  States.  Of  the 
lands  south  of  Lansing,  it  is  now  forty  years  since  the  major 
part  were  sold,  and  the  work  of  improvement  began.  A 
generation  has  passed  away  since  they  were  converted  into 
cultivated  farms,  and  few  if  any  of  the  original  corner  and 
quarter  stakes  now  remain. 

The  corner  and  quarter  stakes  were  often  nothing  but 
green  sticks  driven  into  the  ground.  Stones  might  be  put 
around  or  over  these  if  they  were  handy,  but  often  they 
were  not,  and  the  witness  trees  must  be  relied  upon  after 
the  stake  was  gone.  Too  often  the  first  settlers  were  care- 
less in  fixing  their  lines  with  accuracy  while  monuments  re- 
mained, and  an  irregular  brush  fence,  or  something  equally 
untrustworthy,  may  have  been  relied  upon  to  keep  in  mind 
where  the  blazed  line  once  was.  A  fire  running  through 
this  might  sweep  it  away,  and  if  nothing  was  substituted  in 
its  place,  the  adjoining  proprietors  might  in  a  few  years  be 
found  disputing  over  their  lines,  and  perhaps  rushing  into 
litigation,  as  soon  as  they  had  occasion  to  cultivate  the 
land  along  the  boundary. 

If  now  the  disputing  parties  call  in  a  surveyor,  it  is  not 
likely  that  any  one  summoned  would  doubt  or  question  that 
his  duty  was  to  find,  if  possible,  the  place  of  the  original 
stakes  which  determined  the  boundary  line  between  the 
proprietors.  However  erroneous  may  have  been  the  original 
survey,  the  monuments  that  were  set  must  nevertheless 
govern,  even  though  the  effect  be  to  make  one  half-quarter 
section  ninety  acres  and  the  one  adjoining  seventy;  for 


SURVEYING   LAW  AND   PRACTICE  295 

parties  buy,  or  are  supposed  to  buy,  in  reference  to  these 
monuments,  and  are  entitled  to  what  is  within  their  lines, 
and  no  more,  be  it  more  or  less.  While  the  witness  trees 
remain,  there  can  generally  be  no  difficulty  in  determining 
the  locality  of  the  stakes. 

When  the  witness  trees  are  gone,  so  that  there  is  no 
longer  record  evidence  of  the  monuments,  it  is  remarkable 
how  many  there  are  who  mistake  altogether  the  duty  that 
now  devolves  upon  the  surveyor.  It  is  by  no  means  un- 
common that  we  find  men,  whose  theoretical  education  is 
thought  to  make  them  experts,  who  think  that  when  the 
monuments  are  gone,  the  only  thing  to  be  done  is  to  place 
new  monuments  where  the  old  ones  should  have  been,  and 
would  have  been,  if  placed  correctly.  This  is  a  serious  mis- 
take. The  problem  is  now  the  same  that  it  was  before: 
To  ascertain,  by  the  best  lights  of  which  the  case  admits, 
where  the  original  lines  were.  The  mistake  above  alluded 
to,  is  supposed  to  have  found  expression  in  our  legislation, 
though  it  is  possible  that  the  real  intent  of  the  act  to  which 
we  shall  refer  is  not  what  is  commonly  supposed. 

An  act  passed  in  1869  (Compiled  Laws,  §  593),  amending 
the  laws  respecting  the  duties  and  powers  of  county  sur- 
veyors, after  providing  for  the  case  of  corners  which  can 
be  identified  by  the  original  field  notes  or  other  unques- 
tionable testimony,  directs  as  follows: 

Second.  —  Extinct  interior  section  corners  must  be  re- 
established at  the  intersection  of  two  right  lines  joining 
the  nearest  known  points  on  the  original  section  lines  east 
and  west  and  north  and  south  of  it. 

Third.  —  Any  extinct  quarter-section  corner,  except  on 
fractional  lines,  must  be  re-established  equidistant  and  in  a 
right  line  between  the  section  corners;  in  all  other  cases 
at  its  proportionate  distance  between  the  nearest  original 
corners  on  the  same  line. 

The  corners  thus  determined,  the  surveyors  are  required 
to  perpetuate  by  noting  bearing  trees  when  timber  is  near. 

To  estimate  properly  this  legislation,  we  must  start  with 
the  admitted  and  unquestionable  fact  that  each  purchaser 
from  government  bought  such  land  as  was  within  the 
original  boundaries,  and  unquestionably  owned  it  up  to  the 
time  when  the  monuments  became  extinct.  If  the  monu- 


296  PRACTICAL   SURVEYING 

ment  was  set  for  an  interior  section  corner,  but  did  not 
happen  to  be  "at  the  intersection  of  two  right  lines  joining 
the  nearest  known  points  on  the  original  section  line  east 
and  west  and  north  and  south  of  it,"  it  nevertheless 
determined  the  extent  of  his  possessions,  and  he  gained 
or  lost,  according  as  the  mistake  did  or  did  not  favor 
him. 

It  will  probably  be  admitted  that  no  man  loses  title  to 
his  land  or  any  part  thereof  merely  because  the  evidences 
become  lost  or  uncertain.  It  may  become  more  difficult 
for  him  to  establish  it  as  against  an  adverse  claimant,  but 
theoretically  the  right  remains;  and  it  remains  as  a  poten- 
tial fact  so  long  as  he  can  present  better  evidence  than  any 
other  person.  And  it  may  often  happen  that  notwithstand- 
ing the  loss  of  all  trace  of  a  section  corner  or  quarter  stake, 
there  will  still  be  evidence  from  which  any  surveyor  will  be 
able  to  determine  with  almost  absolute  certainty  where  the 
original  boundary  was  between  two  government  sub- 
divisions. 

There  are  two  senses  in  which  the  word  extinct  may  be 
used  in  this  connection:  One,  the  sense  of  physical  disap- 
pearance ;  the  other,  the  sense  of  loss  of  all  reliable  evidence. 
If  the  statute  speaks  of  extinct  corners  in  the  former  sense, 
it  is  plain  that  a  serious  mistake  was  made  in  supposing  that 
surveyors  could  be  clothed  with  authority  to  establish  new 
corners  by  an  arbitrary  rule  in  such  cases.  As  well  might 
the  statute  declare  that  if  a  man  loses  his  deed,  he  shall  lose 
his  land  altogether. 

But  if  by  extinct  corner  is  meant  one  in  respect  to  the 
actual  location  of  which  all  reliable  evidence  is  lost,  then 
the  following  remarks  are  pertinent: 

1 .  There  would  undoubtedly  be  a  presumption  in  such  a 
case  that  the  corner  was  correctly  fixed  by  the  government 
surveyor  where  the  field  notes  indicated  it  to  be. 

2.  But  this  is  only  a  presumption,  and  may  be  overcome 
by  any  satisfactory  evidence  showing  that  in  fact  it  was 
placed  elsewhere. 

3.  No  statute  can  confer  upon  a  county  surveyor  the 
power  to  "establish"  corners,  and  thereby  bind  the  parties 
concerned.     Nor    is    this    a    question    merely    of    conflict 
between  State  and  federal  law.     It  is  a  question  of  property 


SURVEYING  LAW  AND   PRACTICE  297 

right.  The  original  surveys  must  govern,  and  the  laws 
under  which  they  were  made  must  govern,  because  the  land 
was  bought  in  reference  to  them;  and  any  legislation, 
whether  State  or  federal,  that  should  have  the  effect  to 
change  these,  would  be  inoperative,  because  disturbing 
vested  rights. 

4.  In  any  case  of  disputed  lines,  unless  the  parties  con- 
cerned settle  the  controversy  by  agreement,  the  determina- 
tion of  it  is  necessarily  a  judicial  act,  and  it  must  proceed 
upon  evidence,  and  give  full  opportunity  for  a  hearing. 
No  arbitrary  rules  of  survey  or  of  evidence  can  be  laid 
down  whereby  it  can  be  adjudged. 

The  general  duty  of  the  surveyor  in  such  a  case  is  plain 
enough.  He  is  not  to  assume  that  a  monument  is  lost 
until  after  he  has  thoroughly  sifted  the  evidence  and  found 
himself  unable  to  trace  it.  Even  then  he  should  hesitate 
long  before  doing  anything  to  the  disturbance  of  settled 
possessions.  Occupation,  especially  if  long  continued, 
often  affords  very  satisfactory  evidence  of  the  original 
boundary  when  no  other  is  attainable;  and  the  surveyor 
should  inquire  when  it  originated,  how,  and  why  the  lines 
were  then  located  as  they  were,  and  whether  a  claim  of 
title  has  always  accompanied  the  possession,  and  give  all 
the  facts  due  force  as  evidence.  Unfortunately,  it  is  known 
that  surveyors  sometimes,  in  supposed  obedience  to  the 
State  statute,  disregard  all  evidences  of  occupation  and 
claim  of  title,  and  plunge  whole  neighborhoods  into  quarrels 
and  litigation  by  assuming  to  "establish"  corners  at  points 
with  which  the  previous  occupation  cannot  harmonize. 
It  is  often  the  case  that  where  one  or  more  corners  are  found 
to  be  extinct,  all  parties  concerned  have  acquiesced  in 
lines  which  were  traced  by  the  guidance  of  some  other 
corner  or  landmark,  which  may  or  may  not  have  been  trust- 
worthy; but  to  bring  these  lines  into  discredit  when  the 
people  concerned  do  not  question  them,  not  only  breeds 
trouble  in  the  neighborhood,  but  it  must  often  subject  the 
surveyor  himself  to  annoyance  and  perhaps  discredit, 
since  in  a  legal  controversy  the  law  as  well  as  common  sense 
must  declare  that  a  supposed  boundary  line  long  acquiesced 
in  is  better  evidence  of  where  the  real  line  should  be  than 
any  survey  made  after  the  original  monuments  have  dis- 


298  PRACTICAL  SURVEYING 

appeared.  (Stewart  v.  Carleton,  31  Mich.  Reports,  270; 
Diehl  v.  Zanger,  39  Mich.  Reports,  601.)  And  county  sur- 
veyors, no  more  than  any  others,  can  conclude  parties  by 
their  surveys. 

The  mischiefs  of  overlooking  the  facts  of  possession  most 
often  appear  in  cities  and  villages.  In  towns  the  block  and 
lot  stakes  soon  disappear,  there  are  no  witness  trees  and 
no  monuments  to  govern  except  such  as  have  been  put  in 
their  places,  or  where  their  places  were  supposed  to  be.  The 
streets  are  likely  soon  to  be  marked  off  by  fences,  and  the 
lots  in  a  block  will  be  measured  off  from  these  without 
looking  farther.  Now  it  may  perhaps  be  known  in  a  par- 
ticular case  that  a  certain  monument  still  remaining  was  a 
starting  point  in  the  original  survey  of  the  town  plat;  or  a 
surveyor  settling  in  the  town  may  take  some  central  point 
as  the  point  of  departure  in  his  surveys,  and  assuming  the 
original  plat  to  be  accurate,  he  will  then  undertake  to  find 
all  streets  and  lots  by  course  and  distance  according  to 
the  plat,  measuring  and  estimating  from  his  point  of  de- 
parture. This  procedure  might  unsettle  every  line  and 
every  monument  existing  by  acquiescence  in  the  town;  it 
would  be  very  likely  to  change  the  lines  of  streets,  and  raise 
controversies  everywhere.  Yet  this  is  what  is  sometimes 
done,  the  surveyor  himself  being  the  first  person  to  raise 
the  disturbing  questions. 

Suppose,  for  example,  a  particular  village  street  has  been 
located  by  acquiescence  and  used  for  many  years  and  the 
proprietors  in  a  certain  block  have  laid  off  their  lots  in  refer- 
ence to  this  practical  location.  Two  lot  owners  quarrel, 
and  one  of  them  calls  in  a  surveyor,  that  he  may  make 
sure  his  neighbor  shall  not  get  an  inch  of  land  from  him. 
This  surveyor  undertakes  to  make  his  survey  accurate, 
whether  the  original  was  so  or  not,  and  the  first  result  is, 
he  notifies  the  lot  owners  that  there  is  error  in  the  street 
line,  and  that  all  fences  should  be  moved,  say  one  foot  to 
the  east.  Perhaps  he  goes  on  to  drive  stakes  through  the 
block  according  to  this  conclusion.  Of  course,  if  he  is 
right  in  doing  this,  all  lines  in  the  village  will  be  unsettled; 
but  we  will  limit  our  attention  to  the  single  block.  It  is 
not  likely  that  the  lot  owners  generally  will  allow  the  new 
survey  to  unsettle  their  possessions,  but  there  is  always  a 


SURVEYING  LAW  AND   PRACTICE  299 

probability  of  finding  some  disposed  to  do  so.  We  shall 
then  have  a  lawsuit;  and  with  what  result? 

It  is  a  common  error  that  lines  do  not  become  fixed  by 
acquiescence  in  a  less  time  than  twenty  years,  in  fact,  by 
statute,  road  lines  may  become  conclusively  fixed  in  ten 
years;  and  there  is  no  particular  time  that  should  be  re- 
quired to  conclude  private  owners,  where  it  appears  that 
they  have  accepted  a  particular  line  as  their  boundary, 
and  all  concerned  have  cultivated  and  claimed  up  to  it. 
Public  policy  requires  that  such  lines  be  not  lightly  dis- 
turbed, or  disturbed  at  all,  after  the  lapse  of  any  consider- 
able time.  The  litigant,  therefore,  who  in  such  a  case  pins 
his  faith  on  the  surveyor,  is  likely  to  suffer  for  his  reliance, 
and  the  surveyor  himself  to  be  mortified  by  a  result  that 
seems  to  impeach  his  judgment. 

Of  course  nothing  in  what  has  been  said  can  require  a 
surveyor  to  conceal  his  own  judgment,  or  to  report  the  facts 
one  way  when  he  believes  them  to  be  another.  He  has  no 
right  to  mislead,  and  he  may  rightfully  express  his  opinion 
that  an  original  monument  was  at  one  place,  when  at  the 
same  time  he  is  satisfied  that  acquiescence  has  fixed  the 
rights  of  parties  as  if  it  were  another.  But  he  would  do 
mischief  if  he  were  to  attempt  to  "establish"  monuments 
which  he  knew  would  tend  to  disturb  settled  rights;  the 
farthest  he  has  a  right  to  go,  as  an  officer  of  the  law,  is  to 
express  his  opinion  where  the  monument  should  be,  at  the 
same  time  that  he  imparts  the  information  to  those  who 
employ  him,  and  who  might  otherwise  be  misled,  that  the 
same  authority  that  makes  him  an  officer  and  entrusts  him 
to  make  surveys,  also  allows  parties  to  settle  their  own 
boundary  lines,  and  considers  acquiescence  in  a  particular 
line  or  monument,  for  any  considerable  period,  as  strong  if 
not  conclusive  evidence  of  such  settlement.  The  peace  of 
the  community  absolutely  requires  this  rule.  It  is  not 
long  since,  that  in  one  of  the  leading  cities  of  the  state  an 
attempt  was  made  to  move  houses  two  or  three  rods  into 
a  street,  on  the  ground  that  a  survey,  under  which  the 
street  has  been  located  for  many  years,  had  been  found  on 
a  more  recent  survey  to  be  erroneous. 

From  the  foregoing  it  will  appear  that  the  duty  of  the 
surveyor  where  boundaries  are  in  dispute  must  be  varied 


300  PRACTICAL  SURVEYING 

by  the  circumstances,  i.  He  is  to  search  for  the  original 
monuments,  or  for  the  places  where  they  were  originally 
located,  and  allow  these  to  control.  (If  he  finds  them,  un- 
less he  has  reason  to  believe  that  agreements  of  the  parties, 
express  or  implied,  have  rendered  them  unimportant.)  By 
monuments  in  the  case  of  government  surveys  we  mean  of 
course  the  corner  and  quarter-stakes ;  blazed  lines  or  marked 
trees  on  the  lines  are  not  monuments;  they  are  merely 
guides  or  finger  posts,  if  we  may  use  the  expression,  to  in- 
form us  with  more  or  less  accuracy  where  the  monuments 
may  be  found.  2.  If  the  original  monuments  are  no  longer 
discoverable,  the  question  of  location  becomes  one  of 
evidence  merely.  It  is  merely  idle  for  any  state  statute 
to  direct  a  surveyor  to  locate  or  "establish"  a  corner,  as 
the  place  of  the  original  monument,  according  to  some 
inflexible  rule.  The  surveyor,  on  the  other  hand,  must  in- 
quire into  all  the  facts,  giving  due  prominence  to  the  acts 
of  the  parties  concerned,  and  always  keeping  in  mind, 
first,  that  neither  his  opinion  nor  his  survey  can  be  conclusive 
upon  parties  concerned;  and,  second,  that  courts  and  juries 
may  be  required  to  follow  after  the  surveyor  over  the  same 
ground,  and  that  it  is  exceedingly  desirable  that  he  govern 
his  actions  by  the  same  lights  and  the  same  rules  that  will 
govern  theirs. 

It  is  always  possible,  when  corners  are  extinct,  that  the 
surveyor  may  usefully  act  as  a  mediator  between  parties, 
and  assist  in  preventing  legal  controversies  by  settling 
doubtful  lines.  Unless  he  is  made  for  this  purpose  an 
arbitrator  by  legal  submission,  the  parties,  of  course,  even 
if  they  consent  to  follow  his  judgment,  cannot,  on  the  basis 
of  mere  consent,  be  compelled  to  do  so;  but  if  he  brings 
about  an  agreement,  and  they  carry  it  into  effect  by  actu- 
ally conforming  their  occupation  to  his  lines,  the  action  will 
conclude  them.  Of  course,  it  is  desirable,  that  all  such 
agreements  be  reduced  to  writing;  but  this  is  not  absolutely 
indispensable  if  they  are  carried  into  effect  without. 

Meander  lines.  —  The  subject  to  which  allusion  will  now 
be  made,  is  taken  up  with  some  reluctance,  because  it  is 
believed  the  general  rules  are  familiar.  Nevertheless,  it  is 
often  found  that  surveyors  misapprehend  them,  or  err  in 
their  application ;  and  as  other  interesting  topics  are  some- 


SURVEYING  LAW  AND   PRACTICE  301 

what  connected  with  this,  a  little  time  devoted  to  it  will 
probably  not  be  altogether  lost.  The  subject  is  that  of 
meander  lines.  These  are  lines  traced  along  the  shores 
of  lakes,  ponds,  and  considerable  rivers,  as  the  measures 
of  quantity  when  sections  are  made  fractional  by  such 
waters.  These  have  determined  the  price  to  be  paid  when 
government  lands  were  bought,  and  perhaps  the  impression 
still  lingers  in  some  minds  that  the  meander  lines  are  bound- 
ary lines,  and  that  all  in  front  of  them  remains  unsold. 

Of  course  this  is  erroneous.  There  was  never  any  doubt 
that,  except  on  the  large  navigable  rivers,  the  boundary  of 
the  owners  of  the  banks  is  the  middle  line  of  the  river; 
and  while  some  courts  have  held  that  this  was  the  rule  on 
all  fresh-water  streams,  large  and  small,  others  have  held  to 
the  doctrine  that  the  title  to  the  bed  of  the  stream  below 
low-water  mark  is  in  the  state,  while  conceding  to  the 
owners  of  the  banks  all  riparian  rights.  The  practical 
difference  is  not  very  important.  In  this  state  the  rule 
that  the  center  line  is  the  boundary  line,  is  applied  to  all 
great  rivers,  including  the  Detroit,  varied  somewhat  by 
the  circumstance  of  there  being  a  distinct  channel  for  navi- 
gation, in  some  cases,  with  the  stream  in  the  main  channel, 
and  also  sometimes  by  the  existence  of  islands. 

The  troublesome  questions  for  surveyors  present  them- 
selves when  the  boundary  line  between  two  contiguous 
estates  is  to  be  continued  from  the  meander  lines  to  the 
center  line  of  the  river.  Of  course,  the  original  survey 
supposes  that  each  purchaser  of  land  on  the  stream  has  a 
water  front  of  the  length  shown  by  the  field  notes;  and  it  is 
presumable  that  he  bought  this  particular  land  because  of 
that  fact.  In  many  cases  it  now  happens  that  the  meander 
line  is  left  some  distance  from  the  shore  by  the  gradual 
change  of  course  of  the  stream,  or  diminution  of  the  flow  of 
water.  Now  the  dividing  line  between  two  government 
subdivisions  might  strike  the  meander  line  at  right  angles, 
or  obliquely;  and,  in  some  cases,  if  it  were  continued  in  the 
same  direction  to  the  center  line  of  the  river,  might  cut  off 
from  the  water  one  of  the  subdivisions  entirely,  or  at  least 
cut  it  off  from  any  privilege  of  navigation,  or  other  valuable 
use  of  the  water,  while  the  other  might  have  a  water  front 
much  greater  than  the  length  of  the  line  crossing  it  at  right 


302  PRACTICAL  SURVEYING 

angles  to  its  side  lines.  The  effect  might  be  that,  of  two 
government  subdivisions  of  equal  size  and  cost,  one  would 
be  of  very  great  value  as  water-front  property,  and  the  other 
comparatively  valueless.  A  rule  which  would  produce  this 
result  would  not  be  just,  and  it  has  not  been  recognized  in 
the  law. 

Nevertheless  it  is  not  easy  to  determine  what  ought  to  be 
the  correct  rule  for  every  case.  If  the  river  has  a  straight 
course,  or  one  nearly  so,  every  man's  equities  will  be  pre- 
served by  this  rule :  Extend  the  line  of  division  between  the 
two  parcels  from  the  meander  line  to  the  center  line  of  the 
river,  as  nearly  as  possible  at  right  angles  to  the  general 
course  of  the  river  at  that  point.  This  will  preserve  to  each 
man  the  water  front  which  the  field  notes  indicated,  except 
as  changes  in  the  water  may  have  affected  it,  and  the  only 
inconvenience  will  be  that  the  division  line  between  differ- 
ent subdivisions  is  likely  to  be  more  or  less  deflected  where 
it  strikes  the  meander  line. 

This  is  the  legal  rule,  and  it  is  not  limited  to  government 
surveys,  but  applies  as  well  to  water  lots  which  appear  as 
such  on  town  plats.  (Bay  City  Gas  Light  Co.,  v.  The 
Industrial  Works,  28  Mich.  Reports,  182.)  It  often  hap- 
pens, therefore,  that  the  lines  of  city  lots  bounded  on  navi- 
gable streams  are  deflected  as  they  strike  the  bank,  or  the 
line  where  the  bank  was  when  the  town  was  first  laid  out. 

When  the  stream  is  very  crooked,  and  especially  if  there 
are  short  bends,  so  the  foregoing  rule  is  incapable  of  strict 
application,  it  is  sometimes  very  difficult  to  determine  what 
should  be  done;  and  in  many  cases  the  surveyor  may  be 
under  the  necessity  of  working  out  a  rule  for  himself.  Of 
course  his  action  cannot  be  conclusive,  but  if  he  adopts  one 
that  follows,  as  nearly  as  the  circumstances  will  admit,  the 
general  rule  above  indicated,  so  as  to  divide  as  near  as  may 
be  the  bed  of  the  stream  among  the  adjoining  owners  in 
proportion  to  their  lines  upon  the  shore,  his  division,  being 
that  of  an  expert,  made  upon  the  ground  and  with  all 
available  lights,  is  likely  to  be  adopted  as  law  for  the  case. 
Judicial  decisions,  into  which  the  surveyor  would  find  it 
prudent  to  look  under  such  circumstances,  will  throw  light 
upon  his  duties  and  may  constitute  a  sufficient  guide  when 
peculiar  cases  arise.  Each  riparian  lot  owner  ought  to 


SURVEYING  LAW  AND   PRACTICE  303 

have  a  line  on  the  legal  boundary,  namely,  the  center  line 
of  the  stream  proportioned  to  the  length  of  his  line  on  the 
shore  and  the  problem  in  each  case  is,  how  this  is  to  be  given 
him.  Alluvion,  when  a  river  changes  its  course,  will  be 
apportioned  by  the  same  rules. 

The  existence  of  islands  in  a  stream  when  the  middle 
line  constitutes  a  boundary,  will  not  affect  the  apportion- 
ment unless  the  islands  were  surveyed  out  as  government 
subdivisions  in  the  original  admeasurement.  Wherever 
that  was  the  case,  the  purchaser  of  the  island  divides  the 
bed  of  the  stream  on  each  side  with  the  owner  of  the  bank, 
and  his  rights  also  extend  above  and  below  the  solid  ground, 
and  are  limited  by  the  peculiarities  of  the  bed  and  the 
channel.  If  an  island  was  not  surveyed  as  a  government 
subdivision  previous  to  the  sale  of  the  bank,  it  is  of  course 
impossible  to  do  this  for  the  purpose  of  government  sale 
afterward,  for  the  reason  that  the  rights  of  the  bank  owners 
are  fixed  by  their  purchase;  when  making  that  they  have 
a  right  to  understand  that  all  land  between  the  meander 
lines,  not  separately  surveyed  and  sold,  will  pass  with  the 
shore  in  the  government  sale;  and  having  this  right,  any- 
thing which  their  purchase  would  include  under  it  cannot  be 
taken  from  them.  It  is  believed,  however,  that  the  federal 
courts  would  not  recognize  the  applicability  of  this  rule 
to  large  navigable  rivers,  such  as  those  uniting  the  great 
lakes. 

On  all  the  little  lakes  of  the  state  which  are  mere  expan- 
sions near  their  mouths  of  the  rivers  passing  through  them 
—  such  as  the  Muskegon,  Pere  Marquette  and  Manistee  — 
the  same  rule  of  bed  ownership  has  been  judicially  applied 
as  applied  to  the  rivers  themselves;  and  the  division  lines 
are  extended  under  the  water  in  the  same  way.  (Rice  v. 
Ruddiman,  10  Mich.,  125.)  If  such  a  lake  were  circular, 
the  lines  would  converge  to  the  center;  if  oblong  or  irregu- 
lar, there  might  be  a  line  in  the  middle  on  which  they 
would  terminate,  whose  course  would  bear  some  relation 
to  that  of  the  shore.  But  it  can  seldom  be  important  to 
follow  the  division  line  very  far  under  the  water,  since  all 
private  rights  are  subject  to  the  public  rights  of  naviga- 
tion and  other  use,  and  any  private  use  of  the  lands  incon- 
sistent with  these  would  be  a  nuisance,  and  punishable  as 


304  PRACTICAL  SURVEYING 

such.  It  is  sometimes  important,  however,  to  run  the 
lines  out  for  considerable  distance,  in  order  to  determine 
where  one  may  lawfully  moor  vessels  or  rafts,  for  the 
winter,  or  cut  ice.  The  ice  crop  that  forms  over  a  man's 
land  of  course  belongs  to  him.  (Lorman  v.  Benson,  8 
Mich.,  18;  Peoples  Ice  Co.  v.  Steamer  Excelsior,  recently 
decided.) 

What  is  said  above  will  show  how  unfounded  is  the  notion, 
which  is  sometimes  advanced,  that  a  riparian  proprietor 
on  a  meandered  river  may  lawfully  raise  the  water  in  the 
stream  without  liability  to  the  proprietors  above,  provided 
he  does  not  raise  it  so  that  it  overflows  the  meander  line. 
The  real  fact  is  that  the  meander  line  has  nothing  to  do 
with  such  a  case,  and  an  action  will  lie  whenever  he  sets 
back  the  water  upon  the  proprietor  above,  whether  the 
overflow  be  below  the  meander  lines  or  above  them. 

As  regards  the  lakes  and  ponds  of  the  state,  one  may 
easily  raise  questions  that  it  would  be  impossible  for  him 
to  settle.  Let  us  suggest  a  few  questions,  some  of  which 
are  easily  answered,  and  some  not: 

1.  To  whom   belongs   the   land   under   these  bodies  of 
water,  where  they  are  not  mere  expansions  of  a  stream 
flowing  through  them? 

2.  What  public  rights  exist  in  them? 

3.  If  there  are  islands  in  them  which  were  not  surveyed 
out  and  sold  by  the  United  States,  can  this  be  done  now? 

Others  will  be  suggested  by  the  answers  given  to  these. 

It  seems  obvious  that  the  rules  of  private  ownership 
which  are  applied  to  rivers  cannot  be  applied  to  the  great 
lakes.  Perhaps  it  should  be  held  that  the  boundary  is  at 
low-water  mark,  but  improvements  beyond  this  would  only 
become  unlawful  when  they  became  nuisances.  Islands 
in  the  great  lakes  would  belong  to  the  United  States  until 
sold,  and  might  be  surveyed  and  measured  for  sale  at  any 
time.  The  right  to  take  fish  in  the  lakes,  or  to  cut  ice,  is 
public,  like  the  right  of  navigation,  but  is  to  be  exercised 
in  such  manner  as  will  not  interfere  with  the  rights  of  shore 
owners.  But  so  far  as  these  public  rights  can  be  the  sub- 
ject of  ownership,  they  belong  to  the  state,  not  to  the 
United  States;  and  so,  it  is  believed,  does  the  bed  of  the 
lake  also.  (Pollord  v.  Hagan,  3  Howard's  U.  S.  Reports.) 


SURVEYING  LAW   AND   PRACTICE  30$ 

But  such  rights  are  not  generally  considered  proper  sub- 
jects of  sale,  but  like  the  right  to  make  use  of  the  public 
highways,  they  are  held  by  the  state  in  trust  for  all  the 
people. 

What  is  said  of  the  large  lakes  may  perhaps  be  said  also 
of  many  of  the  interior  lakes  of  the  state;  such,  for  example, 
as  Houghton,  Higgins,  Cheboygan,  Hurt's,  Mullet,  Whit- 
more,  and  many  others.  But  there  are  many  little  lakes 
or  ponds  which  are  gradually  disappearing,  and  the  shore 
proprietorship  advances  pari  passu  as  the  waters  recede. 
If  these  are  of  any  considerable  size  —  say,  even  a  mile 
across  —  there  may  be  questions  of  conflicting  rights  which 
no  adjudication  hitherto  made  could  settle.  Let  any  sur- 
veyor, for  example,  take  the  case  of  a  pond  of  irregular 
form,  occupying  a  mile  square  or  more  of  territory,  and 
undertake  to  determine  the  rights  of  the  shore  proprietors 
to  its  bed  when  it  shall  totally  disappear,  and  he  will  find 
he  is  in  the  midst  of  problems  such  as  probably  he  has  never 
grappled  with,  or  reflected  upon  before.  But  the  general 
rules  for  the  extension  of  shore  lines,  which  have  already 
been  laid  down,  should  govern  such  cases,  or  at  least  should 
serve  as  guides  in  their  settlement. 

Where  a  pond  is  so  small  as  to  be  included  within  the 
lines  of  a  private  purchase  from  the  government,  it  is  not 
believed  the  public  have  any  rights  in  it  whatever.  Where 
it  is  not  so  included,  it  is  believed  they  have  rights  of  fish- 
ery, rights  to  take  ice  and  water,  and  rights  of  navigation 
for  business  or  pleasure.  This  is  the  common  belief,  and 
probably  the  just  one.  Shore  rights  must  not  be  so  exer- 
cised as  to  disturb  these,  and  the  states  may  pass  all  proper 
laws  for  their  protection.  It  would  be  easy  with  suit- 
able legislation  to  preserve  these  little  bodies  of  water  as 
permanent  places  of  resort  for  the  pleasure  and  recreation 
of  the  people,  and  there  ought  to  be  such  legislation. 

If  the  state  should  be  recognized  as  owner  of  the  beds  of 
these  small  lakes  and  ponds,  it  would  not  be  owner  for  the 
purpose  of  selling.  It  would  be  owner  only  as  trustee  for 
the  public  use;  and  a  sale  would  be  inconsistent  with  the 
right  of  the  bank  owners  to  make  use  of  the  water  in  its 
natural  condition  in  connection  with  their  estates.  Some 
of  them  might  be  made  salable  lands  by  draining;  but 


306  PRACTICAL  SURVEYING 

the  state  could  not  drain,  even  for  this  purpose,  against 
the  will  of  the  shore  owners,  unless  their  rights  were  appro- 
priated and  paid  for. 

Upon  many  questions  that  might  arise  between  the  state 
as  owner  of  the  bed  of  the  little  lake  and  the  shore  owners, 
it  would  be  presumptuous  to  express  an  opinion  now,  and 
fortunately  the  occasion  does  not  require  it. 

I  have  thus  indicated  a  few  of  the  questions  with  which 
surveyors  may  now  and  then  have  occasion  to  deal,  and  to 
which  they  should  bring  good  sense  and  sound  judgment. 
Surveyors  are  not  and  cannot  be  judicial  officers,  but  in  a 
great  many  cases  they  act  in  a  quasi  judicial  capacity  with 
the  acquiescence  of  parties  concerned;  and  it  is  important 
for  them  to  know  by  what  rules  they  are  to  be  guided  in 
the  discharge  of  their  judicial  functions.  What  I  have  said 
cannot  contribute  much  to  their  enlightenment,  but  I 
trust  will  not  be  wholly  without  value.  (Finis.) 

Commenting  upon  the  famous  address  of  Judge  Cooley 
it  is  right  to  inform  the  surveyor  that  in  his  dealings  with 
owners  and  their  attorneys  he  will  find  there  is  a  thing 
known  as  "common  law"  and  another  thing  known  as 
"statute  law."  The  common  law  governs  in  the  majority 
of  states  until  a  statute  is  passed  to  settle  if  possible  un- 
certainties which  lead  to  lawsuits.  These  statutes  are 
passed  by  state  legislatures,  in  which  the  majority  of 
members  usually  are  attorneys  (by  courtesy,  lawyers)  and 
whenever  the  legislature  meets,  one  of  the  first  committees 
appointed  is  a  committee  to  pass  upon  the  constitutionality 
of  all  proposed  legislation.  The  statute  referred  to,  which 
purported  to  enable  surveyors  to  "re-establish"  corners, 
was  the  product  of  a  legislature  in  which  the  usual  per- 
centage of  membership  was  lawyers  and  having  the  cus- 
tomary committee  to  scan  all  proposed  laws.  Such  statutes 
have  been  passed  in  a  number  of  states  but  when  inter- 
ested owners  object  and  take  a  case  to  the  Supreme  Court 
the  statute  is  held  to  violate  the  common  law 'doctrine  of 
rights  of  property  and  to  be  unconstitutional.  The  reasons 
are  well  stated  by  Judge  Cooley.  Yet  many  surveyors 
have  been  compelled  by  attorneys  to  follow  the  letter  of 
the  law,  even  while  these  same  attorneys  knew  how  the 
case  would  go  if  carried  to  the  proper  court.  Many  sur- 


SURVEYING   LAW  AND   PRACTICE  307 

veyors  made  mistakes  through  believing  every  law  on  the 
statute  books  to  be  good  law,  but  it  was  not  customary 
fifty  years  ago  for  legal  points  to  be  touched  upon  in  text- 
books on  surveying. 

In  his  reference  to  lot  lines,  Judge  Cooley  had  particular 
conditions  in  mind.  On  the  points  he  brings  up  there  is 
considerable  room  for  differences  of  opinion.  Rights 
never  run  against  the  public.  That  is,  when  a  man  en- 
croaches upon  a  public  highway  with  a  fence  or  building  the 
public  still  has  every  right  it  was  given  so  possession  in 
such  a  manner  does  not  give  title.  However,  if  the  man 
was  permitted  to  place  any  part  of  his  building  on  public 
property  he  gains  an  easement  thereby  for  that  structure. 
He  cannot  be  compelled  to  remove  it,  without  being  com- 
pensated for  whatever  damages  he  may  thereby  suffer, 
but  the  public  may  declare  its  right  in  the  ground  occupied 
and  forbid  him  making  repairs  so  the  structure  may  remain 
indefinitely.  The  public  may  also,  when  the  structure  is 
razed,  compel  the  new  structure  to  be  placed  where  it  prop- 
erly belongs,  thus  terminating  the  easement.  This  again 
is  governed  by  the  conditions  under  which  the  public  rights 
in  the  highway  were  acquired.  The  street  may  have  been 
given  outright  to  the  public.  It  may  have  been  dedicated 
merely  for  highway  purposes  and  nothing  said  about  rever- 
sion in  case  of  abandonment.  Conditions  of  reversion  may 
have  been  stated  at  the  time  of  dedication.  The  ownership 
of  adjacent  lots  may  be  limited  to  the  exact  boundaries  of 
the  lots  and  the  edge  of  the  highway;  it  may  extend  to  the 
center  line  of  the  highway. 

Assume  a  town  site  to  have  been  carefully  surveyed  and 
well  monumented.  Stakes  in  some  way  became  lost  and 
the  people  who  took  possession  were  too  penurious  to  pay 
five  dollars  or  ten  dollars  for  each  lot  survey.  In  course  of 
time  every  fence  line  and  many  building  lines  are  not  in 
the  position  indicated  by  the  records.  Some  street  im- 
provements are  started  and  the  encroachments  discovered. 
The  encroachment  may  be  a  fence  in  such  a  state  of  dilap- 
idation that  the  necessary  street  grading  causes  it  to  fall, 
whereupon  the  city  seizes  the  property  thus  abandoned. 
The  lot  owner  finds  he  has  lost  a  strip  of  land  and  pro- 
ceeds to  demand  it  from  his  neighbor,  the  trouble  running 


308  PRACTICAL  SURVEYING 

through  the  block,  until  at  the  far  end  a  surplus  is  dis- 
covered in  the  last  lot.  This  owner  proposes  to  fight  for 
the  continued  possession  of  the  surplus.  The  monuments 
are  uncovered  and  a  careful  re-survey  made  with  the 
result  that  it  is  found  every  person  in  the  town  may  have 
all  the  land  his,  or  her,  deeds  called  for  and  the  city  may 
also  have  all  the  width  in  each  street.  Nobody  loses  but 
everyone  is  put  to  considerable  expense  and  annoyance. 
This  is  typical  and  all  the  surveyor  can  do  is  to  show  the 
facts.  There  will  be  plenty  of  lawyers  to  take  either  side 
in  a  controversy  and  depending  upon  the  cleverness  of  the 
lawyers  and  the  wealth,  or  poverty,  of  the  litigants  a  crop 
of  decisions  will  be  given  in  the  lower  courts  which  will 
disgust  intelligent  jurists.  If  the  general  rule  laid  down 
by  Judge  Cooley  be  followed  everyone  should  be  per- 
mitted to  stay  put  where  found,  but  his  rule  referred  to 
carelessness  in  the  original  surveys  and  the  subsequent  dis- 
appearance of  stakes  and  monuments. 

Possession  implies  two  things.  Physical  possession  and 
payment  of  taxes.  If  the  deed  calls  for  Lot  5  in  Block  B, 
the  owner  will  be  taxed  for  Lot  5  in  Block  B.  If  his  fence 
encroaches  on  Lot  6  he  obtains  an  easement  so  long  as  the 
owner  of  Lot  6  does  not  object,  but  the  complaisance  of 
that  owner  gives  the  encroacher  no  title  to  a  part  of  Lot  6 
without  the  payment  of  taxes.  He  has  a  sort  of  squatter 
claim  as  it  were,  modified  by  the  law  of  the  state.  It  is 
under  some  such  rule  that  the  case  of  the  carefully  sur- 
veyed town  site  with  the  owners  too  stingy  to  make  surveys 
must  be  considered.  It  is  not  believable  that  all  the  people 
can  be  compelled  to  get  off  the  land  of  their  neighbors  after 
a  long  undisturbed  possession  of  their  neighbor's  land,  but 
if  they  remove  the  encroachment  they  cannot  again  en- 
croach. Also  if  the  owner  of  a  lot  wishes  to  fully  occupy 
it  he  has  the  right  to  excavate  up  to  his  line  and  if  the  en- 
croachment thereby  falls  the  owner  of  the  encroachment 
cannot  claim  damages,  but  if  the  structure  on  the  property 
he  actually  owns  is  damaged,  then  he  can  stand  in  court. 
The  foregoing  is  again  to  be  modified  by  circumstances, 
all  of  which  become  matters  of  evidence  and  thereby  give 
all  parties  concerned  some  reason  for  taking  their  troubles 
to  court.  If  it  were  the  custom  to  employ  a  board  of 


SURVEYING  LAW  AND  PRACTICE        309 

three  arbitrators  consisting  of  an  intelligent  surveyor,  a 
genuine  lawyer  and  a  real  estate  agent  of  good  reputation 
and  long  experience,  such  a  board  probably  would  settle  all 
disputes  in  exactly  the  way  they  would  be  settled  in  the 
Supreme  Court  of  a  state  where  the  Supreme  Court  is 
indifferent  to  the  dotting  of  an  i  or  the  crossing  of  a  t, 
provided  the  case  has  been  sensibly  presented  to  the  court 
and  substantial  justice  is  desired.  Common  law  is  the 
crystallization  of  the  common  sense  of  the  people  for  un- 
numbered generations.  Statute  law  is  not  always  on  a 
plane  with  common  law,  many  times  representing  the  opin- 
ions of  men  elected  to  serve  the  people,  but  really  repre- 
senting some  special  interest  which  may  be  a  corporation; 
and  may  be  an  exasperated  community. 

The  general  rule  about  length  of  possession,  as  well  as 
the  rule  regarding  supposed  acquiescence,  may  be  often 
capable  of  modification.  Both  cases  fail  when  the  undis- 
turbed possession  or  acquiescence  may  be  the  result  of 
fraud,  or  of  a  mistake  made  many  years  before.  Fraud 
vitiates  all  contracts  and  acquiescence  is  the  performance 
of  a  contract  or  understanding,  a  contract  being  a  mutual 
agreement  enforceable  at  law.  The  surveyor  cannot  there- 
fore be  too  critical  of  lawyers  whose  sole  interest  is  that  of 
their  clients.  The  surveyor's  interest  is  merely  to  ascer- 
tain the  facts  and  report  them.  He  may,  if  a  man  of  con- 
siderable experience,  be  of  great  assistance  in  smoothing 
the  feelings  of  property  owners  and  can  suggest  means  for 
avoiding  actions  at  law.  The  general  criticism  against 
lawyers  is  that,  as  a  rule,  they  are  averse  to  seeing  that  a 
surveyor  is  properly  paid  for  his  work,  while  charging  good 
fees  themselves,  and  they  too  often  try  to  tell  a  surveyor 
how  his  work  should  be  done  and  object  when  he  attempts 
to  show  that  to  recover  a  line  it  may  be  frequently  neces- 
sary to  run  many  lines  and  take  many  days  to  ascertain  the 
facts.  When  a  man  requires  the  services  of  a  physician  or 
surgeon  he  makes  no  attempt  to  secure  bids  and  award  the 
work  to  the  lowest  bidder,  but  rather  he  secures  as  soon  as 
possible  a  man  with  a  good  reputation.  He  acts  similarly 
when  engaging  a  lawyer,  proceeding  upon  the  theory  that 
expense  cannot  be  spared  when  he  is  compelled  to  go  into 
court.  When  the  employment  of  an  engineer  or  surveyor 


310  PRACTICAL  SURVEYING 

is  necessary  attempts  are  made  to  employ  the  man  who  will 
work  for  the  least  money  and  too  often  the  selection  is  left 
to  the  lawyer,  who  is  generally  inclined  to  employ  the 
cheapest  workman.  Young  men  are  preferred  as  a  rule 
because  it  is  supposed  they  can  get  over  more  land  in  a  given 
time  than  older  men.  Surveyors  in  the  course  of  years 
acquire  a  fund  of  valuable  experience,  and  many  times 
lawyers  are  employed  on  property  line  disputes  only  once 
or  twice  in  a  lifetime.  It  is  irksome  to  competent  sur- 
veyors to  be  employed  under  the  direction  of  such  men, 
hence  the  friction  between  the  two  professions. 

Occasionally  a  surveyor  becomes  so  interested  in  the 
legal  phases  of  his  work  that  he  studies  law  and  is  admitted 
to  practice.  John  Cassan  Wait  of  New  York  City  is  an 
example  of  this  class,  for  he  made  a  good  reputation  as  a 
civil  engineer  and  then  became  a  lawyer  whose  dictum  in 
such  matters  is  considered  conclusive.  William  E.  Kern, 
Attorney-at-law  and  Civil  Engineer,  wrote  an  article  on  the 
legal  side  of  surveys  which  was  printed  in  Engineering  News, 
Vol.  48,  and  the  author  obtained  permission  from  his  widow 
to  reprint  the  article  in  this  book. 

A  BRIEF  DISCUSSION  OF  THE  LAW  OF  BOUNDARY 
SURVEYS 

By  WILLIAM  E.  KERN,  C.  E. 
Copyright  1902,  by  William  E.  Kern. 

A  surveyor  may  enhance  his  reputation  for  piety  by 
being  able  to  cite  from  the  Bible,  Deut.  XXVII:  17  and 
XIX:  14;  Job  XXIV:  2;  Prov.  XXII:  28  and  XXIII:  10, 
the  verses  mentioned  referring  to  the  sinfulness  of  removing 
land  marks  and  the  penalty  therefor.  Josephus  has 
something  to  say  on  the  same  subject,  and  in  his  first 
volume  of  "Antiquities  of  the  Jews,"  Chapter  II,  he  inti- 
mates that  the  first  murderer  was  also  the  first  surveyor. 
In  the  I2th  book  of  the  "^neid,"  Virgil  recites  an  unusual 
use  of  a  monument.  The  esteem  of  the  ancients  is  shown 
by  their  provision  of  a  special  deity,  Terminus,  to  preside 
over  boundary  matters,  all  land  marks  of  stone  being  re- 
garded as  monuments  to  his  godship. 


SURVEYING  LAW  AND   PRACTICE  311 

So  much  for  ancient  history.  The  modern  surveyor  is 
often  called  upon  to  decide  boundary  disputes,  where  the 
best  results  would  flow  from  a  union  of  his  efforts  with  those 
of  a  counsellor  at  law.  In  view  of  this  condition  and  the 
many  questions  arising  in  the  practice  of  a  surveyor,  some 
systematic  knowledge  of  the  subject  on  his  part  would  seem 
desirable. 

Several  works  upon  surveying  contain  a  number  of  syllabi 
of  cases  involving  boundary  disputes.  These  syllabi  are 
condensed  statements  of  the  law  applicable  to  a  particular 
combination  of  circumstances;  and  have  limited  utility 
for  the  purpose  for  which  they  are  inserted  in  such  text- 
books. All  laymen  must  be  cautioned  against  making  a 
general  application  of  an  abstract  statement  of  law,  as 
determined  in  some  special  case.  To  illustrate  this,  the 
following  is  quoted  from  a  textbook  on  surveying:  "Seventy 
acres  lying  and  being  in  the  southwest  corner  of  a  section 
is  a  good  description,  and  the  land  will  be  in  a  square. 
W.  v.  R.  2  Ham.  Ohio  327."  This  is  sufficient  reference 
for  a  lawyer,  as  he  would  look  up  all  the  facts;  but  for  a 
surveyor's  enlightenment  it  should  be  explained  that  the 
description  read  "Seventy  acres  lying,  and  being  in  the 
southwest  corner  of,"  a  certain  section,  etc.,  "of  the  lands 
sold  at  Steubenville, "  and  nothing  further.  The  piece  in 
question  was  part  of  a  larger  lot,  and  had  never  been  marked ; 
nor  was  there  anything  to  indicate  the  intention  as  to  its 
shape.  The  court  disagreed  with  the  views  on  both  sides 
and  held  it  must  be  surveyed  as  a  square. 

The  term  "boundary"  is  used  in  two  senses,  and  may  be 
denned  as:  A  series  of  lines  forming  the  perimeter  of  a 
specific  tract  of  land;  or,  an  object,  parcel  of  land,  or  body 
of  water  contiguous  to  a  specific  tract  of  land.  "Land" 
in  legal  parlance  usually  means  a  portion  of  the  earth's 
surface,  whether  dry  or  covered  with  water.  Every  tract 
of  land  with  which  we  may  be  called  upon  to  deal,  can  be 
considered  as  having  at  some  prior  time  formed  part  of  a 
larger  tract.  Incident  to  the  severance  was  the  creation  of 
a  new  boundary  along  the  line  of  division.  Every  bound- 
ary in  dispute  may  be  conceived  as  having  had  such  an 
origin.  After  its  inception,  however,  neighbors  on  either 
side  may,  by  friendly  agreements  between  each  other;  or 


312  PRACTICAL  SURVEYING 

by  long-continued  hostile  encroachments  by  one,  with 
neglect  to  assert  rights  by  the  other;  or  by  other  acts; 
affect  the  legal  status  of  the  line  so  as  to  alter  its  original 
position. 

The  data  for  determining  the  legal  status  of  a  boundary 
may  be  divided  into  two  classes:  (i)  Circumstances  con- 
nected with  the  severance  from  a  parent  tract  at  the  time 
the  line  in  question  first  became  a  boundary,  and  (2)  con- 
duct of  neighbors  on  either  side  of  the  line,  which  may 
affect  their  relative  status  thereto. 

EXTENT   OF   GRANT 

The  first  class  above  referred  to  may  be  designated  as 
Extent  of  Grant.  In  discussing  this  subdivision,  it  is 
almost  superfluous  to  say  that  the  rules  stated  may  be 
limited  in  their  application  in  specific  cases  by  the  princi- 
ples falling  under  the  second  class.  In  locating  a  boundary 
as  determined  by  extent  of  grant,  we  must  seek  the  inten- 
tion of  the  parties  to  the  primal  conveyance. 

Let  us  suppose  that  A,  being  the  owner  of  a  large  plot 
of  ground,  divides  it  into  two  portions  by  a  line  XY  run- 
ning north  and  south,  and  sells  all  east  of  XY  to  B,  and  sub- 
sequently all  on  the  west  to  M.  B  then  sells  to  C,  the 
latter  to  D,  and  D  to  E;  M  conveys  to  N.  At  a  time  when 
N  is  owning  on  one  side  and  E  on  the  other,  it  becomes 
necessary  to  'define  the  line  XY.  This  must  be  based 
upon  the  description  used  in  A  to  B,  it  being  the  one  which 
created  the  line  X  Y.  No  conveyance  in  the  chain  of  A  to 
N  can  affect  the  rights  acquired  by  E  through  the  con- 
veyance of  A  to  B.  The  intermediate  deeds  should  be 
examined,  however,  as  they  frequently  throw  light  upon 
the  original  intention,  as  for  example:  If  M  took  in  1821, 
N  in  1860,  B  in  1820,  C  in  1840,  D  in  1860,  and  E  in  1880, 
the  monuments  called  for  in  1820  may  be  gone  in  1900, 
but  a  surveyor  in  1820  might  have  found  vestiges  and 
restored  them,  or  created  new  marks  in  correct  positions; 
and  another  surveyor  in  1860,  finding  the  marks  of  1820, 
may  have  made  an  accurate  mathematical  description  and 
set  two  or  more  imperishable  monuments. 

It  may  be  urged  that  the  original  deed  cannot  be  found, 


SURVEYING  LAW  AND  PRACTICE  313 

or  is  too  indefinite  to  be  of  any  service.  The  attempt,  how- 
ever, ought  always  be  made;  if  it  fails  we  must  do  the 
next  best  thing.  In  a  recent  case  the  mesne  conveyances 
of  a  plot  within  the  limits  of  a  large  city  differed  about 
100  ft.,  in  frontage  on  a  street.  An  examination  of  the 
original  deed  disclosed  a  call  for  the  edge  of  a  fast  land  along 
the  side  of  a  swamp;  a  city  plan,  showing  topography, 
fixed  the  position  of  the  water  line  and  solved  the  problem. 
The  intention  of  the  parties  is  generally  to  be  gathered 
from  the  description  in  the  instrument  of  conveyance. 
Description  we  will  define  as  a  statement  designed  to  identi- 
fy a  specific  tract  of  land.  Every  specific  tract  of  land  is 
included  between  lines  of  definite  length,  making  equally 
definite  angles  with  each  other  and  with  the  meridian,  and 
containing  a  definite  area;  and  any  angle  point  may  be 
conceived  as  located  at  a  definite  distance  and  direction 
from  some  well-known  and  fixed  point.  Some  specific 
tracts  are  marked  at  angle  points  or  elsewhere  by  natural 
or  artificial  monuments;  some  are  known  by  particular  or 
distinguishing  names.  A  complete  description  would 
contain  all  of  these  elements  so  far  as  they  exist. 

MONUMENTS 

Monuments  are  objects  established  or  used  to  indicate 
the  boundary  of  a  specific  tract  of  land.  They  may  be 
natural  or  artificial,  the  distinction  being  too  obvious  to 
require  definition.  Under  this  head  are  included  varieties 
of  land,  as  swamps,  meadows,  forests,  pasture,  etc.;  bodies 
of  water  and  water  courses;  hillocks,  cliffs,  trees,  rocks, 
buildings,  walls,  fences,  stones,  stakes,  pits,  etc. 

Ambiguity  in  a  description  may  arise  from  conflict 
between  elements  of  a  different  sort,  or  between  those  of 
the  same  sort.  Monuments  may  disagree  with  the  mathe- 
matical description,  or  with  each  other,  or  be  missing;  the 
dimensions  may  disagree  with  each  other  or  with  the  area. 
When  discrepancies  arise,  the  cardinal  rule  to  follow  is, 
effectuate  the  intention  of  the  original  parties.  Where  there 
is  sufficient  evidence  to  explain  the  conflict  and  point  out 
the  mistake,  the  line  must  be  run  upon  the  reformed  data 
whether  it  cause  the  rejection  of  a  monument  or  dimension. 


314  PRACTICAL  SURVEYING 

It  most  frequently  happens,  however,  that  no  such  evidence 
is  at  hand,  in  which  event  the  following  rules  are  to  be 
applied : 

Monuments  are  the  best  evidence  of  location  of  a  bound- 
ary, and  natural  monuments  are  to  be  preferred  to  all  other 
forms  of  description.  This  rule  is  so  important  that  a  list 
of  cases  is  added  containing  one  from  nearly  every  state 
in  the  Union.  A  perusal  of  any  one  case  will  show  the  reason 
for  the  rule,  and  probably,  in  addition,  refer  to  other  cases 
in  the  same  state  which  may  be  looked  up,  as  the  scope  of 
this  article  will  not  permit  of  stating  all  the  authority  for 
the  rules  stated.* 

This  rule,  of  course,  applied  only  where  the  monuments 
were  adopted  as  such,  either  by  mention  in  the  description 
or  by  reference  to  or  use  of  a  survey  wherein  they  are  estab- 
lished or  used  (Beckman  v.  Davidson,  162  Mass.  347). 
In  such  cases  it  is  not  necessary  that  they  be  seen  by  the 
grantor  to  bind  him.  No  matter  how  exactly  the  survey 
closes  or  agrees  with  the  content,  if  the  monuments  do  not 
agree  therewith,  the  former  falls.  If  a  line  is  marked  on 
the  ground  in  any  manner,  though  described  as  straight,  it 
must  follow  the  breaks  in  the  marked  line  if  the  latter  is 
not  straight.  An  instance  of  this  kind  frequently  occurs 
in  running  up  a  creek;  several  reaches  are  often  combined 
in  one  course;  in  such  case  the  line  must  follow  the  water. 
This  occurred  in  Spring  v.  Hewston,  52  Cal.  442. 

In  Esmond  v.  Tarbox,  7  Me.  61,  a  survey  was  made  by 

*  Ayers  v.  Watson,  137  U.  S.  584;  Wright  v.  Wright,  34  Ala.  194; 
Stoll  v.  Beecher,  94  Cal.  i;  Nichols  v.  Turney,  15  Conn.  101;  Nivin  v. 
Stevens,  5  Harr.  (Del.)  272;  Andreu  v.  Watkins,  26  Flo.  390;  Harris  v. 
Hull,  70  Ga.  831;  Bolden  v.  Sherman,  no  111.  418;  Shepherd  v.  Nave, 
125  Ind.  226;  Yocum  v.  Haskins,  81  Iowa  436;  Bruce  v.  Morgan,  I  B. 
Mon.  (Ky.)  26;  Lebeau  v.  Bergeron,  14  La.  Ann.  494;  Oxton  v.  Groves, 
68  Me.  371 ;  Wood  v.  Ramsey,  71  Md.  9;  Woodward  v.  Nims,  130  Mass. 
70;  Bruckner  v.  Lawrence,  i  Doug.  (Mich.)  19;  Newman  v.  Foster, 
3  How.  (Miss.)  383;  Harding  v.  Wright,  119  Mo.  i;  Johnsons.  Preston, 
9  Neb.  474;  Cunningham  v.  Curtis,  57  N.  H.  158;  Kalbfleisch  v.  Oil  Co., 
43  N.  J.  L.  259;  Thayer  v.  Finton,  108  N.  Y.  394;  Redmond  v.  Stepp, 
100  N.  C.  212;  Hare  v.  Harris,  14  Ohio  529;  Anderson  v.  McCormick, 
1 8  Oreg.  301;  Morse  v.  Rollins,  121  Pa.  537;  Faulwood  v.  Graham, 
I.  Rich.  L.  (S.  C.)  491;  Lewis  v.  Oakley,  19  Heis  (Tenn.)  483;  Wyatt 
v.  Foster,  79  Tex.  413;  Grand  Trunk  Co.  v.  Dyer,  49  Vt.  74;  Coles  v. 
Wooding,  2  P.  &  H.  (Virg.)  189;  Teass  v.  St.  Albans,  38  W.  Va.  i; 
Miner  v.  Brader,  65  Wis.  537. 


SURVEYING  LAW  AND   PRACTICE  315 

one  surveyor  and  marked  with  monuments,  and  a  plan 
made  by  another;  the  deed  recited  the  latter  only.  A  dis- 
agreement between  the  two  having  been  found,  it  was 
held  that  the  monuments  governed.  In  Wilson  v.  Bass, 
6  Tenn.  no,  a  course  called  for  a  known  point,  and  then 
crossing  a  river  at  200  poles  continued  213^  poles  to  its 
terminus.  It  was  found  that  the  true  distance  to  the  river 
was  213  poles.  It  was  held  that  as  the  parties  had  agreed 
that  it  was  200  poles  to  the  river,  the  line  must  extend  13 
poles  beyond  the  same.  In  Frey  v.  Baker,  7  Ky.  L.  Rep. 
663,  testator  divided  a  tract  into  two  pieces,  and  described 
the  dividing  line  as  beginning  at  a  point  at  a  fence,  and 
running  thence  west  along  the  fence.  It  was  shown  that 
the  fence  did  not  run  east  and  west,  and  that  testator  was 
ignorant  of  that  fact.  It  was  held  that  the  fence  and  not  an 
imaginary  line  due  west  was  the  proper  line. 

Wendell  v.  Jackson,  8  Wend.  183,  is  an  early  but  inter- 
esting New  York  case.  A  patent  had  been  issued  for  a 
tract  described  as  beginning  at  the  easterly 
corner  of  township  No.  20,  etc.,  and  proceeding 
by  various  courses  and  distances,  Fig.  211. 
The  third  course  ran  8*50°  £153  chains  to  the 
side  of  Schroon  Lake,  with  content  as  3500 
.acres.  The  actual  distance  to  the  lake  on  the 
third  course  FG  was  less  than  one-half  of  the 
distance  stated.  A  survey  for  an  adjacent 
tract,  made  seven  days  later,  began  at  a  clump 
of  rocks  on  the  side  of  Schroon  Lake,  at  a  corner 
of  the  preceding  patent,  thence  followed  the  same  around 
N  50°  W  153  chains,  S  40°  W  105.5  chains  S  31°  15'.  E 
330  chains  to  corner  of  town  20,  etc.  It  was  necessary  to 
locate  the  first  patent  to  determine  the  title  to  the  land 
northwest  of  EF  and  southeast  of  the  line  //.  The  town- 
ship corner  used  as  a  beginning  was  well  marked  by  a  stake 
and  stones,  and  was  at  a  point  in  a  swamp  difficult  of  access. 
It  was  argued  on  one  side  that  the  call  for  the  beginning  at 
D  should  be  disregarded,  that  the  second  survey  following 
the  first  so  closely,  and  having  been  made  so  short  a  time 
before,  should  throw  light  on  the  latter,  and  that  the  first 
should  start  at  A  on  the  side  of  the  lake,  and,  reversing  the 
courses,  run  backwards  until  arriving  at  A.  This  would 


316  PRACTICAL  SURVEYING 

locate  the  land  at  AJIHK.  It  was  held  by  a  divided  court 
of  14  to  5  that  the  beginning  D  being  a  known  point  must 
be  used  and  the  line  run  to  the  side  of  the  lake  at  A ,  thence 
to  beginning,  or  to  B.  The  point  being,  the  supremacy  of 
the  call  for  a  natural  monument  on  the  "side  of  a  lake." 
This  case  might  have  been  decided  otherwise  if  many 
members  of  the  court  had  ever  had  any  practical  experience 
at  surveying.  It  is  quite  probable  that  the  first  survey  was 
begun  at  a  point  H,  and  the  line  between  H  and  D  not  run. 
owing  to  the  difficulty  of  getting  over  a  bog.  The  adding  of 
an  estimated  distance  HD  was  forgotten,  and  AK  and  HK 
calculated  to  make  the  figure  close. 

On  the  other  hand,  in  White  v.  Leming,  it  was  held  that 
where  rejecting  the  call  for  one  monument  would  reconcile 
the  other  parts  of  the  deed  and  leave  enough  to  identify 
the  land,  the  rule  was  not  applicable. 

As  between  natural  and  artificial  monuments  the  former 
will  have  the  greatest  weight.  Stakes,  being  perishable  and 
so  easily  moved,  are  regarded  as  very  inferior  marks;  in 
one  case,  Huffman  v.  Walker,  83-N.  C.  411,  the  identifica- 
tion of  the  location  was  considered  impossible.  Where  a 
monument  called  for  is  lost,  its  original  position  must  be 
ascertained  if  possible.  This  may  be  done  by  the  testi- 
mony of  witnesses  as  to  its  former  location,  by  indications 
on  the  ground,  or,  in  the  absence  of  these,  by  such  methods 
as  appear  to  be  the  nearest  to  certainty. 

RESTORING  LOST   SECTION   CORNERS 

In  restoring  lost  section  or  quarter  corners,  various  and 
conflicting  rules  have  been  stated  by  the  land  office  and 
courts.  In  order  to  understand  how  to  restore  corners  of 
the  public  land  surveys,  some  knowledge  of  the  method  of 
making  the  original  surveys  should  first  be  had.  The 
details  of  the  latter  may  be  learned  from  several  textbooks, 
or  better  still,  from  the  manual  of  instructions  issued  by 
the  government  to  deputy  land  surveyors.  It  may  be 
summarized  briefly  as  follows:  In  several  different  parts  of 
the  western  country,  initial  points  have  been  established 
from  which  base  lines  have  been  run  towards  all  four  points 
of  the  compass.  The  north  and  south  line,  being  straight 


SURVEYING  LAW  AND   PRACTICE  317 

its  entire  length,  is  referred  to  as  the  principal  meridian 
for  its  section  of  the  domain.  The  east  and  west  line  is 
called  the  principal  base,  and  is  likewise  straight  its  full 
distance.  From  points  on  the  meridian,  at  intervals  which 
are  multiples  of  six  miles,  lines  are  extended  east  and  west 
and  are  known  as  standard  parallels.  On  the  base  and 
parallels,  similar  multiples  of  six  miles  are  laid  off,  and  from 
their  termini  guide  meridians  are  extended  north  to  the 
adjacent  parallels.  The  blocks  thus  formed  are  subdivided 
into  six-mile  rectangles  by  lines  due  east  and  west  and  north 
and  south,  these  blocks  being  called  townships.  Sections 
of  one  mile  square  are  next  surveyed  by  beginning  at  a 
point  on  the  south  boundary  of  a  township  one  mile  west 
of  its  south  east  corner,  and  running  slightly  west  of  north, 
so  that  each  mile  point  shall  be  one  mile  from  the  east 
boundary  of  the  township,  all  error  being  thrown  towards 
the  north  and  west.  Each  half-mile  point  is  marked  by 
a  "quarter  corner." 

In  1885  the  General  Land  Office  issued  a  pamphlet  en- 
titled "Restoration  of  Lost  and  Obliterated  Corners." 
This  seems  to  be  a  very  good  article  except,  perhaps,  in  the 
method  it  directs  for  finding  the  center  of  a  section.  Where 
a  government  corner  is  missing,  and  after  diligent  inquiry 
its  former  position  cannot  be  ascertained,  the  surveyor 
should  proceed  as  follows:  If  the  point  was  on  a  base, 
parallel  or  meridian,  it  must  be  restored  to  a  position  on  a 
line  between  the  nearest  corners  on  the  same  parallel  or 
base,  and  at  a  distance  which  is  a  proportional  part  of  the 
whole  distance  between  the  known  corners.  To  illustrate, 
suppose  that  the  intersection  of  a  meridian  with  a  parallel 
is  marked,  and  the  nearest  corner  south  is  a  section  corner 
two  miles  away.  Three  marks  are  missing.  The  distance 
measures  159.2  chains,  and  the  field  notes  of  the  original 
survey  show  the  distances  going  north  to  be  40  chains  to 
first-quarter  corner,  40  to  the  section  corner,  40  to  the  next 
quarter,  and  39  chains  to  close.  The  excess  of  0.2  must  be 
equated  all  along  the  line  so  that  the  distances  become  40.05, 
40.05,  40.05,  39.05,  respectively,  and  corners  are  set  at  these 
distances  on  a  line  between  the  two  known  corners. 

In  the  same  manner,  points  on  the  boundaries  of  a*  town- 
ship should  be  set  at  equated  distances,  and  on  straight 


318  PRACTICAL  SURVEYING 

lines  between  the  nearest  marks  on  the  same  boundary, 
except  in  the  case  of  the  township  corners  when  on  a  parallel 
or  base.  Owing  to  the  system  of  completing  each  township 
by  itself,  its  corners  are  not  necessarily  on  straight  lines. 
To  re-locate  a  township  corner,  not  on  a  base  or  parallel, 
a  trial  line  should  be  run  between  the  nearest  corners  north 
and  south  and  the  distance  measured  and  compared  with 
the  field  notes.  A  point  is  then  marked  on  the  trial  line 
at  the  equated  distance.  From  this  point  the  distances  to 
nearest  points  east  and  west  are  to  be  measured,  added  and 
compared  with  the  field  notes,  and  an  equated  distance  as- 
certained. The  point  first  fixed  is  to  be  moved  to  the  east 
or  west  to  suit  its  equated  distance  in  that  direction. 

The  same  course  is  to  be  followed  in  re-locating  section 
corners.  Let  us  suppose  (Fig.  212)  the  section  corner  A 
.and  quarter  corners  B  and  C  are  lost  and  we  wish  to  re-lo- 
^  cate  A.  D,  E,  F  and  G  are  found 

marked.  A  trial  line  is  run  from 
E  to  G  and  the  distance  meas- 
ured, temporary  corners  being 
marked  at  A  and  B,  B  being 

£  placed  at  39  chains  from  E  and  A 

f. 


C 

, — Measured 


II 1 54 


<L   40  chains  from  B,  these  being  the 


j     distances  in  the  original  field  notes. 

The  whole  length  having  been  re- 
L^is  ported  as  119  chains  is  found  to 

F  measure  but  118.70  a  deficiency  of 

.30  chains.  This  being  equated 

between  EB,  BA  and  AG,  B  must  be  moved  N  o.io 
chains  and  A  placed  0.20  chains  north  of  their  first  tem- 
porary positions,  D  to  F  is  then  measured  and  found 
111.54  chains,  as  compared  with  no  chains  on  the  official 
plat.  This  is  an  excess  of  1.54  chains,  and  duly  equated 
would  make  DC  30.42,  CA  and  AF  each  40.56.  A  is  then 
to  be  set  40.56  chains  from  F  and  39.90  chains  from  G. 
The  quarter  corners  are  to  be  set  on  a  straight  line  be- 
tween their  section  corners,  hence  B  is  to  be  located  on 
a  line  from  A  to  E  39.90  chains  from  A,  and  quarter 
corner  C  on  a  line  between  D  and  A  and  40.56  from  A. 


SURVEYING  LAW  AND   PRACTICE  319 

FINDING   CENTER  OF   SECTION 

Two  methods  have  been  mentioned  for  finding  the  center 
of  a  section.  The  pamphlet  above  referred  to  suggests 
that  it  be  placed  at  the  intersection  of  lines  run  from  the 
opposite  quarter  corners.  Another  method  was  to  place 
it  at  a  point  equidistant  from  the  quarter  corners,  except 
in  the  north  and  west  tiers  of  sections,  where  it  would  be 
located  as  described  above  for  section  corners,  using  the 
quarters  as  a  basis.  From  a  legal  standpoint  the  following 
would  probably  be  preferable: 

When  completely  marked,  the  exterior  of  a  section  has 
eight  monuments.  From  carelessness,  these  frequently 
disagree  with  the  field  notes.  Quarters  are  often  out  of 
line  and  nearer  one  corner  than  the  other.  In  spite  of 
these  defects,  when  set,  the  marks  must  not  be  altered  in 
position.  Again,  fractional  subdivisions  are  described  as 
"quarter  sections,"  indicating  the  idea  of  quantity;  hence 
a  quarter  section  should  be  surveyed  as  one-fourth  of  the 
area  of  the  section  as  physically  defined.  This  should  be 
done  by  first  making  a  careful  survey  of  the  entire  section 
and  calculating  its  actual  area.  A  point  should  then  be 
fixed  so  that  lines  run  from  it  to  the  quarter  corners  east, 
west  and  south  would  include  exactly  one-fourth  of  the 
entire  calculated  area.  This  would  determine  a  boundary 
for  the  S  E  and  S  W  quarters.  A  second 
point  on  the  line  between  the  E  and  W  A 
quarters  at  such  position,  that  a  line 
drawn  from  it  to  the  quarter  corner  north 
would  divide  the  north  half  equally, 
should  then  be  fixed. 

The  section  might  then  have  two  cen- 
ters, as  it  were,  but  usually  one  point 
would  answer  the  requirements.  Fig.  213  FlG 

will  illustrate  a  possible  condition.  Sup- 
pose the  field  notes  show  a  section  80  by  80,  with  quarter 
corners  all  40  chains  from  the  section  corners,  and  a  meas- 
urement shows  an  area  of  640  acres,  but  the  quarter  corners 
at  incorrect  distances  as  indicated.  A  point  /  is  found  by 
calculation,  so  that  by  joining  H,  /,  FI  and  DI  the  S  E  and 
S  W  quarters  are  each  1 60  acres.  A  second  point  /  is 


320  PRACTICAL  SURVEYING 

found,  so  that  by  joining  BJ  the  N  E  and  N  W  quarters  are 
equal  to  each  other  and  the  other  two.  The  patentee  of  the 
N  W  quarters  cannot  complain  at  his  irregular  lines,  as  he 
receives  more  land  than  by  running  straight  lines  between 
the  quarters  corners,  as  indicated  by  the  land-office  circular. 

FAULTY  DESCRIPTION 

Where  a  description  by  course  and  distance  does  not 
close  in  itself,  courses  may  be  reversed  if  necessary,  and 
clearly  erroneous,  as  for  example:  A  description  read, 
"Beginning  at  the  center  of  a  railroad  at  its  intersection 
with  the  B  road,  thence  S  20°  W  along  said  center  of  said 
railroad  150  ft.,  thence  N  70°  W  50  ft.  to  a  point,  thence 
N  20°  E  150  ft.  to  a  point,  thence  N  70°  W  50  ft.  to  begin- 
ning."  Here  the  fourth  course  was  plainly  in  error,  and 
was  read  as  S  70°  E,  as  there  was  evidence  to  show  that 
the  fourth  and  not  the  second  course  was  erroneous.  A 
similar  error  occurred  in  Brown  et  al  v.  Hage,  21  How.  320, 
where  a  patent  read,  "Beginning  at  a  sycamore  standing 
on  the  edge  of  the  Shanadoah  River,  and  extending  thence 
down  the  said  river  (N  48°  W  200  ch.),  etc.,  etc."  The 
title  to  a  large  area  depended  on  the  retention  or  rejection 
of  the  words  in  the  parenthesis,  and  the  court  said  it  was 
clear  that  to  go  down  the  river  would  not  be  northwesterly, 
it  being  a  matter  of  common  knowledge  that  the  river  ran 
in  the  opposite  direction  at  this  point, 
and  the  words  were  rejected. 

Where,  however,  a  natural  monument 
is  called  for  by  mistake,  such  calls  will  be 
rejected.  Thus  in  Land  Co.  v.  Thompson, 
83  Tex.  169,  several  grants  began  "on  the 
side  of  Devil's  River,"  under  a  mistaken 
apprehension  as  to  its  real  position,  no 
FIG.  214  survey  having  been  made  on  the  ground. 

It  was  held  that  the  surveys  must  be  made 
by  course  and  distance,  recourse  having  been  had  to  other 
established  marks  on  the  east,  and  the  land  between  the 
true  and  false  position  of  the  river  was  excluded  from  the 
grants  calling  for  it.  The  two  positions  are  shown  in  the 
annexed  sketch,  Fig.  214. 


SURVEYING  LAW  AND   PRACTICE  321 

Patents  calling  for  fractional  lots  along  rivers  do  not 
take  the  land  between  the  meander  survey  line  and  the 
river  in  Nebraska,  but  this  state  seems  to  differ  from  the 
others,  in  decisions  on  the  point.  In  Jefferis  v.  Land  Co., 
134  U.  S.  178  and  Ayers  v.  Watson,  the  contrary  was  held, 
and  United  States  statutes  of  1796,  1800,  1805,  1820,  1832, 
and  Sec.  2395,  Rev.  Stat.  U.  S.  cited  to  sustain  the  doctrine 
that  the  land  between  the  meander  lines  and  river  does  pass 
with  the  fractional  lots,  even  though  the  river  be  navigable. 
Quantity  is  usually  subordinated  to  both  course  and  dis- 
tance as  being  the  least  certain  element.  Occasionally  it 
may  govern,  however.  In  one  case  a  deed  called  for  the 
side  of  a  certain  highway,  which  at  the  time  of  the  dis- 
pute was  obliterated.  The  rear  line  was  defined  on  the 
ground.  Several  positions  for  the  lost  highway  were  found. 
It  was  held  that  the  one  agreeing  most  nearly  with  the  area 
described  should  be  preferred,  as  none  agreed  with  the 
course  and  distance. 

Calls  for  adjoining  surveys  or  tracts  must  be  observed. 
This  question  most  frequently  arises  in  patents  from  state 
governments.  In  .the  same  manner  a  plan  or  plat  when 
referred  to  must  be  considered  a  part  of  the  description. 
But  where  the  plan  conflicts  with  monuments  of  an  actual 
survey  on  the  ground,  the  monuments  govern;  unless  the 
survey  was  subsequent  to  the  plan,  when,  in  such  case,  the 
latter  controls. 

The  term  more  or  less  is  construed  as  importing  inexact- 
ness in  all  quantities  to  which  it  is  annexed.  It  might  seem 
ridiculous  to  describe  a  lot  in  feet,  inches  and  thirty- 
seconds  of  an  inch,  and  add  to  each  length  "more  or  less," 
but  if  adjacent  properties  or  other  monuments  are  called 
for,  and  the  distances  to  such  marks  disagree  with  those  of 
the  description,  the  statement  of  the  indefiniteness  of  the 
distances  in  the  description  becomes  of  some  value,  but,  at 
the  most,  of  very  slight  account,  as  the  monuments  would 
control  as  well  without  the  words  more  or  less  as  with 
them.  Cases  where  they  are  of  importance  will  seldom 
arise. 

When  course  and  distance  disagree  with  each  other,  that 
will  prevail  which  under  all  the  circumstances  and  evidence 
appears  to  be  the  most  certain. 


322  PRACTICAL  SURVEYING 

The  cases  generally  arise  in  this  way:  The  opposite  ends 
of  two  adjoining  courses  being  agreed  upon  as  definitely 
determined  and  located,  and  one  course  being  extended  to 
meet  the  other  reversed  to  run  from  its  opposite  terminus, 
the  point  of  intersection  is  found  to  disagree  with  the  dis- 
tances. The  general  rule  is  that  courses  prevail  over  dis- 
tances. Thus,  in  Curtis  v.  Aaronson,  49  N.  J.  L.  68,  1886, 
both  parties  agreed  until  the  i6th  course  was  run.  This 
was  described  as  S  24°  E  29  chains,  thence  865°  15'  W 
151.5  chains  to  pine  on  east  side  of  Shoal  Branch.  The 
position  of  this  pine  was  not  disputed,  nor  was  the  initial 
point  of  the  i6th  course;  but  by  running  the  latter  S  24°  E 
to  intersect  the  iyth  course  extended  N  65°  15'  E  from  the 
pine,  the  i6th  distance  would  be  lengthened  to  96.5  chains 
in  lieu  of  29  chains  as  stated.  The  surveyors  in  re-running 
the  entire  survey  where  the  lines  were  not  disputed,  found 
the  courses  all  approximately  correct,  but  the  distances 
very  erroneous.  This  fact,  among  others  determined  the 
court  to  hold  that  the  disputed  courses  must  be  inter- 
sected and  the  disputed  distances  ignored. 

On  the  other  hand,  in  a  somewhat  similar  case  in  Ken- 
tucky in  1821,  the  discrepancy  between  accuracy  of  courses 
as  compared  with  distances  did  not  appear,  and  the  court 
directed  the  distances  to  be  maintained,  as  by  inter- 
secting the  courses  the  distances  would  be  very  greatly 
distorted. 

Areas  are  regarded  as  the  least  certain  of  the  elements  of 
a  description.  So  where  courses  and  distances  do  not  in- 
clude the  same  area  as  that  stated,  the  latter  must  yield, 
unless  intention  is  clearly  expressed  that  an  area  of  specific 
quantity  shall  be  conveyed,  in  which  latter  case  the  area 
and  not  the  distances  should  control  in  construing  the 
description  of  the  land.  Sanders  v.  Golding,  45  Iowa 

463- 

Plats  or  maps  referred  to  in  a  deed  are  to  be  considered 
as  a  part  of  the  description.  The  following  illustrates 
their  status:  In  Vance  v.  Fore,  24  Cal.  435,  a  deed  referred 
to  another  deed,  previously  made,  for  a  description  of  the 
premises  granted,  and  also  to  a  map  not  contemporaneous 
with  the  last  deed.  The  description  in  the  older  deed  called 
for  no  monument  at  the  initial  point,  nor  could  any  be 


SURVEYING  LAW  AND   PRACTICE  323 

identified  with  accuracy.  The  first  course  terminated  at 
"the  base  of  the  mountain,  thence  running  at  right  angles, 
following  down  the  base  of  the  mountains."  The  char- 
acter of  the  country  was  such  that  witnesses  might  reason- 
ably differ  in  their  location  of  the  lines.  The  map  referred 
to  showed  all  the  natural  and  artificial  monuments  found  on 
the  ground,  such  as  streams,  buildings  and  roads.  The 
descriptions  as  set  forth  in  the  old  deed  and  on  the  map 
conflicted.  It  was  held  that  the  description  shown  on  the 
plat  was  the  most  certain  under  the  circumstances,  and  the 
least  likely  to  be  affected  with  mistakes,  hence  to  be  fol- 
lowed in  making  the  re-survey. 

There  is  an  apparent  conflict  in  the  cases -as  to  the  rela- 
tive precedence  of  plats,  and  the  elements  of  course,  dis- 
tance, etc.,  but,  as  explained  before,  all  cases  must  be 
considered  from  the  standpoint  of  principle  involved.  The 
principle  in  this  case  being  that  that  which  is  shown  to  be 
the  most  certain  will  prevail. 

In  Heaton  v.  Hodges,  14  Me.  66,  no  survey  could  be  ascer- 
tained to  have  been  made,  and  the  monuments  shown  on 
the  plan  could  not  be  identified;  certain  distances  on  the 
plan  disagreed  with  measurements  of  established  lines. 
It  was  held  that  the  disputed  lines  were  to  be  first  scaled 
from  the  plat  and  the  lengths  so  ascertained,  altered  in 
such  proportion  as  the  length  of  the  accepted  lines  bore  to 
their  dimensions  on  the  plat. 

In  Lampe  v.  Kennedy,  45  Wis.  23,  several  deeds  were 
made  from  a  plat.  The  courses  and  distances  did  not  agree 
with  the  plat,  but  the  latter  prevailed.  In  the  same  man- 
ner, plats  were  held  to  control  quantity,  in  109  111.  46, 
and  Hathaway  v.  Power,  5  Hill  (N.  Y.)  453. 

In  Beaty  v.  Robertson,  130  Ind.  589,  it  was  said  that  where 
the  plat  and  field  notes  of  a  government  survey  conflicted, 
the  former  showed  the  lines  as  fixed  by  the  surveyor-gen- 
eral, and  were  those  by  which  the  land  was, sold,  hence  to  be 
taken. 

The  last  method  of  describing  land  is  by  a  particular 
name.  In  earlier  times  this  was  necessarily  the  most 
common.  In  such  cases  disputed  boundaries  are  estab- 
lished by  the  evidence  of  witnesses  familiar  with  the  oldest 
physical  location  of  the  lines,  or  the  admissions  of  inter- 


324  PRACTICAL  SURVEYING 

ested  parties.  This  class  would  include,  however,  descrip- 
tion by  lot  and  block  number  as  per  a  plat.  Town-site 
plats  usually  show  in  addition  to  the  dimensions  of  the  lots, 
stone  monuments  set  at  intersection  of  specified  offset 
range  lines  at  street  corners.  The  monuments  are  fre- 
quently found  to  be  inaccurately  set,  and  various  expedients 
have  been  adopted  by  later  surveyors  to  eliminate  the 
error.  In  some  cases  the  total  error  of  each  block  has  been 
thrown  in  the  adjacent  street;  in  still  others,  the  streets 
have  been  made  the  proper  width  and  the  error  equated 
through  the  lots.  Probably  the  best  method  by  analogy 
with  the  decisions  on  lost  government  corners,  would  be 
to  measure  the  distance  between  monuments  and  compare 
the  same  with  the  distance  between  the  same  points  as 
shown  on  the  plat.  The  difference  should  then  be  equated 
for  every  foot  alike,  whether  streets  or  lots. 


STANDARD   OF  MEASUREMENTS 

Congress  has  never  exercised  its  constitutional  right  to 
fix  a  standard  of  linear  measurement,  but  an  official  standard 
having  been  adopted  by  the  United  States  surveyors,  a  so- 
called  United  States  standard  foot  has  arisen,  and  several 
states  have  enacted  statutes  adopting  this  standard,  which, 
in  the  absence  of  Congressional  action,  is  lawful  for  the 
respective  states  so  adopting  them,  and  all  standards  used 
in  contravention  of  such  state  standards  are  illegal  in  these 
states.  This  gives  rise  to  some  important  questions,  which 
will  now  be  considered. 

Where  a  piece  of  ground  is  conveyed  and  the  descrip- 
tion used  is  an  illegal  standard,  that  standard  must  be  used 
to  establish  its  boundaries;  but  if  it  cannot  be  shown  that 
the  illegal  standard  was  adopted  by  the  parties,  then  the 
construction  must  be  based  upon  the  legal  standard  of  the 
state.  To  make  the  case  more  concrete:  If  A  owns  500  ft. 
front  of  unimproved  land  and  sells  100  ft.  from  one  end,  this 
must  be  measured  in  legal  standard;  but  if  he  owns  one 
lot,  described  as  either  20  or  100  ft.,  but  by  an  illegal 
standard,  and  the  further  words  are  added,  "being  the 
same  premises, "  etc.,  then  he  would  take  the  same  premises, 


SURVEYING  LAW  AND   PRACTICE  325 

or  all  that  his  grantor  took  in  the  recited  conveyance, 
whether  more  or  less.  In  other  words,  it  is  a  question  of 
intention;  if  both  parties  meant  the  same  thing,  then  that 
is  what  they  contracted  for,  or  else  nothing. 

PROPERTY  ABUTTING  ON  ROADS  AND  STREETS 

Where  land  is  conveyed  by  side  of  a  road,  street  or  alley, 
the  usual  practice  is  that,  by  implied  grant,  half  the  road  is 
conveyed  with  the  abutting  lot.  In  some  states  the  fee 
of  the  streets  is  in  the  public;  in  such  cases  the  abutter 
stops  at  the  side  of  the  road.  In  a  few  isolated  cases  the 
law  has  been  held  to  be  that  a  conveyance  did  not  pass 
title  to  the  center  by  implication." 

Thus,  in  Union  Cemetery  v.  Robinson,  5  Wh.  (Pa.)  18, 
the  general  principle  that  a  conveyance  passed  a  fee  to  the 
center  of  the  road  was  not  doubted,  but  it  was  said  that  as 
in  the  case  at  bar  the  street  described  was  only  on  paper, 
and  unopened,  and  further,  that  the  description  being  in 
feet,  inches  and  fractions  of  an  inch,  much  stress  should  be 
laid  upon  the  minuteness  of  the  measurements,  as  showing 
an  intention  to  limit  the  conveyance  to  the  side  of  the  road 
or  street.  This  case  has  very  properly  been  overruled  in 
the  same  and  other  states,  both  as  respects  the  importance 
of  the  refinement  of  measurements  and  as  to  the  fact  that 
the  street  was  unopened.  The  leading  case  on  the  subject 
being  Paul  v.  Carver,  26  Pa.  223.  The  principle  that  a 
conveyance  along  a  road  carries  the  boundary  to  the  cen- 
ter of  that  road  has  been  followed  in  practically  every  state 
of  the  Union,  although  there  have  been  instances  where 
courts  have  been  swayed  by  exceptional  circumstances  to 
hold  that  an  intention  was  manifest  to  exclude  the  road. 
The  words  "by  the  side  of"  have  been  given  that  effect; 
also  a  grant  of  right  of  way  over  the  strip  included  in  the 
street.  The  following  from  Paul  v.  Carver  is  a  forcible 
statement  of  the  reason  of  the  rule:  "The  rule  had  its 
origin  in  a  regard  to  the  nature  of  the  grant.  Where  land 
is  laid  out  in  town  lots,  with  streets  and  alleys,  the  owner 
receives  full  consideration  for  the  streets  and  alleys  in  the 
increased  value  of  the  lots.  The  understanding  always  is 
that  houses  may  be  erected  fronting  on  the  streets  with 


PRACTICAL  SURVEYING 

windows  and  doors,  doorsteps  and  vaults.  If  a  right  of 
property  in  the  streets  might  under  any  circumstances  be 
exercised  by  the  grantor,  he  might  deprive  his  grantee  of 
the  means  of  entry  into  or  exit  from  his  house,  and  of  all 
the  enjoyments  of  light  and  air,  and  might  thereby  deprive 
him  of  the  means  of  deriving  any  benefit  from  his  pur- 
chase." 

Where,  however,  the  original  line  between  two  adjacent 
tracts  existed  before  the  road  was  laid  out,  and  was  not 
coincident  with  the  center  line  of  the  road;  then  it,  and  not 
such  center  line,  is  the  boundary. 

Where  the  conveyance  is  along  the  margin  of  a  river  or 
other  water  way,  the  decisions  are  conflicting,  and  no  pre- 
cise rule  can  be  stated.  In  the  majority  of  the  states, 
however,  a  conveyance  by  the  side  of  a  tidal  or  navigable 
river  or  sea  carries  the  fee  to  high-water  mark,  and  on  un- 
navigable  rivers  and  ponds  to  their  center.  The  side  lines 
are  projected  at  right  angles  with  the  thread  of  the  stream, 
unless  otherwise  provided  for  in  the  deed. 


BOUNDARIES  FIXED   BY  AGREEMENT   OR  A  QUIESCENCE 

Boundaries  may  be  fixed  by  agreement  of  the  owners 
affected,  and  in  such  cases  bind  those  who  take  from  them. 
But  if  either  party  was  led  to  agree  by  a  mistake  in  a  meas- 
urement made  for  the  purpose,  he  may  repudiate  his  agree- 
ment if  he  does  so  within  a  short  time  after  discovery  of 
the  mistake.  Coon  v.  Smith,  29  N.  Y.  392.  An  agreement 
to  fix  a  boundary  where  it  is  indefinite  need  not  be  in  writ- 
ing. Turner  v.  Baker,  64  Mo.  218. 

In  each  state  of  the  Union  statutes  exist  prescribing  a 
length  of  time  during  which  various  suits  may  be  brought, 
and  upon  failure  to  bring  such  suit  the  cause  is  lost.  The 
periods  of  these  several  "Statutes  of  Limitations,"  vary  in 
the  several  states.  A  boundary,  though  erroneous,  if 
acquiesced  in  by  the  parties  for  a  term  beyond  that  of  the 
appropriate  statute,  becomes  fixed,  and  as  suit  could  not  be 
brought  to  rectify  it,  it  should  be  surveyed  as  maintained. 

Estoppel  is  a  legal  principle  by  which  one  who  has  led 
another  to  believe  that  certain  conditions  arc  true,  is  after- 


SURVEYING  LAW  AND   PRACTICE  327 

wards  precluded  from  showing  such  conditions  are  false, 
where  the  innocent  party  would  thereby  be  injured.  So 
boundaries  may  become  fixed  by  estoppel.  Thus,  in 
Sheridan  v.  Barret,  one  led  another  to  believe  that  their 
boundary  was  in  a  certain  location,  and  to  build  a  wall  in 
a  position  based  upon  that  fact.  The  first  was  compelled 
to  adopt  the  boundary  which  he  had  falsely  represented  as 
being  correct.  See  also  New  York  Co.  v.  Gardner,  25  S.  W. 
737;  Jordan  v.  Deaton,  23  Ark.  704. 

In  re-marking  a  boundary,  we  are  first  to  satisfy  ourselves 
that  no  line  has  been  acquiesced  in,  agreed  upon  or  main- 
tained by  adverse  possession  for  the  statutory  period.  If 
not,  the  deeds  are  to  be  examined  and  the  line  traced  on  the 
ground  as  originally  run ;  if  no  vestiges  of  the  latter  remain, 
then  the  elements  of  description  are  to  be  observed  in  the 
order  named,  bearing  in  mind  that  that  which  is  most 
certain  will  prevail  over  the  less  certain.  (Finis.) 

GOVERNMENT  SURVEYS 

When  a  district  was  established  for  the  disposal  of  public 
lands  an  Initial  Point  —  of  which  there  are  31  in  the  United 
States  —  was  selected  within  the  district.  Through  this 
point  was  run  a  north  and  south  line  called  the  Principal 
Meridian;  and  an  east  and  west  line  called  the  Principal 
Base. 

At  intervals  of  24  miles  on  the  Principal  Meridian, 
measured  from  the  initial  point,  Standard  Parallel  or  Cor- 
rection Lines  run  east  and  west  parallel  with  the  Principal 
Base.  Beginning  from  the  Initial  Point  they  are  known  as 
1st,  2nd,  3rd,  etc.,  Standard  Parallel  North  (or  South). 

From  the  Principal  Base  and  Standard  Parallels,  true 
north  lines  called  guide  Meridians  run  at  intervals  of  24 
miles.  Thus  the  land  is  divided  into  ."checks"  approx- 
imately 24  miles  square. 

The  standard  parallels  were  called  Correction  Lines  be- 
cause the  Guide  Meridians  being  true  Meridians  converged 
on  account  of  the  spherical  shape  of  the  earth,  so  a  "check" 
is  narrower  on  the  north  than  on  the  south.  On  each 
standard  parallel  two  stakes  not  far  apart  are  set  for  each 
guide  meridian,  which  has  caused  many  lawsuits  because  the 


328  PRACTICAL  SURVEYING 

first  settlers,  their  attorneys  and  surveyors  knew  little  or 
nothing  of  astronomy  and  geodesy. 

Each  check  is  divided  into  townships  measuring  six 
miles  north  and  south  and  six  miles  wide  on  the  south  end. 
The  north  and  south  township  boundaries  are  parallel  but 
the  east  and  west  boundaries  being  true  meridians  con- 
verge, hence  there  are  again  two  stakes  marking  meridians 
where  parallels  intersect  them. 

Each  township  is  divided  into  sections  one  mile  square 
(approximately)  and  in  surveying  the  sections  (subdivid- 
ing a  township),  monuments  were  supposed  to  be  set  every 
half  mile.  The  corners  are  called  "section"  corners  and 
the  half-mile  corners  are  called  "Quarter  Corners, "  because 
they  divide  the  section  into  quarters.  No  stakes  were 
set  in  the  interior  of  a  section. 

The  original  surveys  were  made  by  contract,  all  contracts 
being  awarded  to  the  lowest  bidders.  Many  ignorant  men 
obtained  contracts  and  so  much  trouble  arose  that  after  a 
time  farcical  examinations  were  held  to  license  surveyors. 
Political  influences  vitiated  this  attempt  to  improve  matters. 
The  business  of  making  government  surveys  assumed  large 
proportions  and  many  frauds  were  perpetrated  by  dishonest 
contractors,  some  of  whom  turned  in  field  notes  for  surveys 
that  were  not  made. 

The  intelligent  men  in  the  land  department  tried  to  check 
fraud  by  having  each  survey  examined  before  paying  the 
contract  price.  The  work  of  examining  surveys  was  let 
also  by  contract  to  the  lowest  bidders,  with  results  any  man 
of  common  sense  could  have  foretold.  At  the  present  time 
examiners  hold  office  for  life  with  a  fair  salary  and  are 
selected  by  means  of  competitive  civil  service  examinations. 

Cheap  work  was  done;  dishonest  work  was  done;  cor- 
ners were  seldom  made  of  permanent  material;  early 
settlers  were  careless;  many  surveyors  who  made  the  first 
re-surveys  were  very  ignorant;  state  legislatures  un- 
fortunately thought  corners  could  be  "re-established";  the 
government  altered  methods  a  number  of  times;  on  all 
surveys  prior  to  1864  an  inaccurate  method  was  used  to 
indicate  courses  on  true  lines ;  these  were  a  few  of  the  many 
causes  for  trouble  over  "government  lines,"  and  the  Land 
Department  was  compelled  to  issue  instructions  to  guide 


SURVEYING  LAW  AND  PRACTICE        329 

surveyors  in  making  re-surveys  in  states  in  which  the  land 
was  ' '  sectionized . ' ' 

The  pamphlet  is  revised  from  time  to  time  and  may  be 
procured  free  of  cost  from  the  General  Land  Office,  Wash- 
ington, D.  C.  The  title  is  "Restoration  of  Lost  or  Obliter- 
ated Corners  and  Subdivisions  of  Sections."  It  should 
be  owned  and  carefully  studied  by  every  land  surveyor. 

The  government  "  Manual  of' Surveying  Instructions  for 
the  Survey  of  the  Public  Lands  of  the  United  States  and 
Private  Land  Claims,"  is  supplied  by  the  Government 
Printing  Office,  Washington,  D.  C.,  for  seventy-five  cents, 
postpaid.  Instructions  for  surveying  and  marking  mining 
claims  may  be  obtained  from  the  Surveyor-General  for  the 
district  in  which  the  claims  lie. 

The  student  after  reading  this  chapter  is  ready  to  appre- 
ciate the  statement  that  most  of  the  trouble  over  bound- 
ary lines  is  caused  by  the  ignorance  of  the  men  who  make 
the  first  re-surveys.  They  are  disposed  to  follow  field 
notes  rather  than  monuments;  or  possession  when  monu- 
ments cannot  be  found.  Field  notes  and  maps  are  often  in 
error,  when  compared  with  modern  standards  of  accuracy, 
but  nevertheless  must  be  used  as  a  guide  in  re-tracement 
surveys,  and  they  always  stand  in  court  until  evidence  is 
introduced  to  destroy  their  credibility.  The  surveyor  must 
refuse  to  accept  work  from  clients,  or  their  attorneys,  who 
attempt  to  dictate  as  to  how  a  survey  must  be  conducted. 
A  single  line  cannot  fix  the  location  of  an  obliterated  monu- 
ment. 

To  subdivide  a  section  all  the  exterior  lines  must  be  re- 
traced and  correct  bearings  and  lengths  obtained.  Then, 
and  not  until  then,  smaller  parcels  may  be  cut  out  and  safely 
rnonumented.  The  plat  and  field  notes  should  show  this. 


CHAPTER   VIII 

ENGINEERING   SURVEYING 

Surveyors  and  civil  engineers  use  the  same  instruments 
and  in  schools  where  surveying  is  taught  as  a  branch  of 
mathematics  instruction  is  given  in  the  use  of  instruments 
and  the  computations  connected  therewith.  From  this 
point  surveying  is  divided  into  : 

1.  Land  surveying. 

2.  Engineering  surveying. 

Land  surveying,  strictly,  is  concerned  only  with  the  divid- 
ing of  land;  to  determine  areas  and  to  re-locate  missing 
monuments  and  boundaries.  Leveling  is  really  engineering 
work  but  it  is  a  part  of  the  work  performed  by  all  land 
surveyors  because  they  are  surveyors. 

Engineering  surveying  consists  in  making  surveys  for 
railways,  highways,  canals,  ditches,  and  the  setting  out 
of  work,  such  as  bridges,  dams,  large  buildings,  tunnels, 
etc.,  together  with  the  computation  of  quantities  of  mate- 
rials. Surveys  for  the  purpose  of  making  contour  maps, 
selecting  the  gradients  for  roads,  surveying  for  the  location 
of  sewers  and  computing  the  amount  of  earthwork  falls 
naturally  to  local  surveyors  who  specialize  in  land  survey- 
ing, because  such  work  may  be  done  by  anyone  familiar 
with  the  use  of  instruments,  surveying  methods  and  draft- 
ing. Mining  surveying  is  engineering  work  which  may  be 
equally  well  done  by  all  qualified  land  surveyors.  Surveys 
made  for  the  purpose  of  preparing  good  maps  belong  both 
to  the  land  surveyor  and  the  civil  engineer.  In  this  chapter 
it  is  proposed  to  present  some  fundamental  information 
relative  to  good  practice  in  engineering  surveying. 

MINING   SURVEYS 

Surveys  made  to  determine  the  boundaries  of  mining 
claims  on  the  surface  of  the  ground  are  called  surface  sur- 
veys. No  further  instructions  are  required  for  such  work 

33° 


ENGINEERING   SURVEYING  331 

than  have  been  given  for  land  surveys.  The  work  is  done 
with  a  transit  and  steel  tape  with  the  greatest  possible 
accuracy.  When  the  location  of  a  mining  claim  is  made  it 
must  be  done  in  accordance  with  instructions  issued  by 
the  Commissioner  of  the  General  Land  Office,  Washington, 
D.  C,  and  the  local  rules  and  regulations  of  the  United 
States  Surveyor  General  within  whose  district  the  claim 
is  located. 

Underground  surveys  are  made  to  show  the  workings 
of  mines.  The  top  of  the  shaft,  or  the  mouth  of  the  tunnel 
where  either  appears  on  the  surface,  is  located  and  tied  to 
some  known  corner  or  permanent  object  on  the  surface. 
A  meridian  line  is  chosen  and  set  out  on  the  surface  for 
some  distance  and  the  ends  marked  with  good  stakes,  so  a 
long  backsight  may  be  obtained.  If  a  tunnel  is  to  be  sur- 
veyed, a  point  on  this  line  is  set  opposite  the  mouth  of 
the  tunnel  and  the  line  of  the  tunnel  is  run  out,  and  tied 
to  the  meridian.  The  line  is  carried  through  the  work- 
ings by  backsight  and  foresight,  all  angles  being  double 
centered  and  all  tack  points  being  set  by  double  centering. 
When  a  survey  is  made  on  the  surface,  it  is  possible  to  run 
around  the  land  and  make  a  closed  survey,  thereby  obtain- 
ing a  check.  When  the  survey  is  run  underground  this 
opportunity  to  make  a  check  does  not  occur,  except  when 
a  point  may  be  projected  up  through  a  shaft,  or  be  carried 
through  another  tunnel  to  the  surface.  The  greater  num- 
ber of  mining  surveys,  however,  take  the  lines  well  under- 
ground and  unless  much  care  is  used  a  mistake  may  be 
made  in  setting  down  R  for  Z,,  or  vice  versa,  and  a  map 
made  from  such  incorrect  notes  will  be  considerably  in 
error. 

Double  centering  is  always  advisable,  but  if  the  transit 
is  in  perfect  adjustment  and  the  surveyor  is  very  careful, 
it  is  well  to  run  the  line  in  with  azimuths  first.  Then 
follow  with  a  carefully  run  double  centered  line.  On 
account  of  the  presence  of  ores  or  of  tracks  and  metal  tools 
the  needle  cannot  check  underground  surveys.  In  coal 
mines  the  needle  may  sometimes  be  used  to  advantage  in 
the  approximate  location  of  rooms  for  placing  same  on 
working  maps,  but  the  cases  are  few  when  the  needle  may 
be  used. 


33 2  PRACTICAL  SURVEYING 

The  preservation  of  points  is  a  vexing  question  in  mine 
surveys.  When  the  ground  is  squeezing  and  sinking  it  is 
impossible  to  preserve  points  for  any  length  of  time  and 
much  of  the  work  of  a  mining  surveyor  is  concerned  with 
the  preservation  of  lines.  Usually  it  is  best  to  drive  nails 
in  the  timber  over  the  tunnels,  these  nails  having  in  them 
holes  through  which  a  plumb-bob  string  is  tied.  Flat- 
headed  horseshoe  nails  with  holes  drilled  in  the  heads  are 
commonly  used.  To  set  up  a  transit  it  is  necessary  to  have 
points  underneath  on  the  ground  so  temporary  points  are 
used.  These  temporary  points  are  often  made  of  pointed 
nails  driven  through  a  piece  of  lead  made  in  the  shape  of  a 
shallow  cone  and  weighing  several  pounds.  The  point  of 
the  nail  is  at  the  top.  A  plumb-bob  is  suspended  from  the 
nail  in  the  roof  and  the  temporary  hub  placed  on  the  ground 
so  the  point  of  the  plumb-bob  is  directly  above  the  point 
of  the  nail.  The  bob  is  then  hung  on  one  side  and  the 
transit  placed  over  the  point.  Backsights  and  foresights 
are  taken  on  the  nails  in  the  roof.  On  account  of  strong 
currents  of  air  in  mine  workings  all  lines  should  be  as  thin 
as  possible  and  plumb-bobs  should  weigh  not  less  than  two 
pounds.  If  they  swing  too  much  they  should  be  immersed 
in  buckets  of  oil  with  the  lower  end  free  of  the  bottom  of  the 
bucket. 

Various  forms  of  plumb-bobs  are  made  for  use  in  mines 
having  lamps  in  the  shank,  so  sights  may  be  taken  on  the 
flame.  The  writer  found  in  his  experience  that  the  best 
sight  is  a  candle  about  two  inches  or  less  in  length.  When 
lighted  the  candle  is  to  some  degree  transparent  and  the 
wick  showing  dimly  through  the  luminous  wax  or  paraffine 
makes  a  very  good  point  on  which  to  sight.  These  lights 
may  be  set  on  the  ground  by  using  the  suspended  plumb-bob 
in  the  manner  mentioned  for  setting  temporary  transit 
points.  For  reading  the  verniers  it  was  formerly  custom- 
ary to  have  tallow  candles,  but  today  small  pocket  electric 
lamps  are  used,  many  instrument  makers  now  supplying 
reading  lamps  made  especially  for  use  in  mines  and  tunnels. 

The  methods  described  are  those  used  in  all  ordinary 
tunnel  work,  the  driving  of  very  costly  and  important 
tunnels  for  railways  being  work  that  is  entrusted  only  to 
men  having  good  experience  in  this  particular  specialty. 


ENGINEERING   SURVEYING  333 

In  mining  work  and  in  the  driving  of  ordinary  tunnels  the 
surveyor  is  interrupted  often  by  the  passing  of  cars,  laborers, 
etc.,  and  as  he  is  not  permitted  to  interrupt  the  regular 
work  by  his  operations,  they  have  the  right  of  way.  This 
calls  for  the  exercise  of  considerable  patience  and  necessi- 
tates rapid  work.  Mistakes,  however,  are  not  forgiven, 
interrupted  work  being  considered  no  excuse  for  mistakes. 
When  the  surveyor  can  choose  his  own  time  to  do  the  work 
he  works  on  Sunday,  provided  the  mine  shuts  down  on  that 
day,  something  seldom  done,  or  he  works  when  the  mine  is 
shut  down,  which  is  a  frequent  occurrence  in  some  lines  of 
mining,  especially  in  a  dull  season.  Usually,  however,  a 
mine  is  run  night  and  day  and  the  surveyor  is  the  least 
considered  of  all  the  laborers,  but  his  work  nevertheless 
must  be  kept  up. 

Underground  surveys  are  made  for  the  purpose  of  keeping 
up  the  office  maps  and  also  for  guiding  the  miners.  When 
done  to  keep  up  the  maps  a  line  is  run  down  the  tunnels, 
and  stakes  set  opposite  openings  in  the  face,  or  opposite 
entrances  to  rooms,  at  regular  stations,  50  to  100  ft.  apart. 
From  these  stakes  approximate  measurements  (to  the  near- 
est half  foot)  are  made  forward  and  back  on  line  to  the 
edges  and  a  measurement  taken  normal  to  the  line  in  to  the 
face  at  each  end  and  at  the  station.  If  closer  results  are 
wanted  closer  measurements  are  made.  The  mine  maps, 
however,  are  on  a  scale  that  will  hardly  show  a  measure- 
ment of  less  than  2  ft.  anything  less  being  estimated. 
When  the  survey  is  made  for  the  purpose  of  guiding  the 
miners,  it  is  necessary  to  set  three  points  on  a  line  pointing 
the  direction  in  which  the  work  is  to  proceed.  The  fore- 
man, or  boss,  then  hangs  three  plumb-bobs  to  these  points 
and  uses  them  for  sighting  purposes.  He  will  carry  his 
work  in  quite  a  distance,  setting  points  ahead  from  which 
to  project  his  line,  when  he  gets  too  far  away  to  see  the 
three  lines  plainly  that  were  left  by  the  surveyor.  When 
he  feels  there  is  danger  that  he  may  be  working  to  one 
side  the  surveyor  is  sent  for  to  set  three  more  points,  one 
close  in  to  the  face. 

Careful  work  is  all  that  is  necessary  in  ordinary  under- 
ground work,  much  care  being  necessary  to  prevent  injury 
to  person  and  instrument.  The  best  mining  surveyors 


334  PRACTICAL  SURVEYING 

are  men  who  have  worked  as  miners  before  studying  sur- 
veying. 

The  greatest  work  of  a  mining  surveyor  is  the  carrying 
of  a  line  from  the  surface  to  the  bottom  of  a  shaft.  Per- 
haps a  shaft  is  being  sunk  in  a  certain  place  and  it  is  in- 
tended to  make  an  upraise  to  complete  the  job,  or  an  up- 
raise is  to  be  started  to  reach  a  certain  point  on  the  surface. 
The  traverse  cannot  be  closed  until  the  shaft,  or  upraise, 
is  finished,  and  a  small  error  in  running  the  lines  may  cost 
thousands  of  dollars. 

First  the  point  on  the  surface  is  located  and  a  line  run 
carefully  to  the  shaft,  two  points  being  set  at  the  shaft, 
one  on  either  side  and  carefully  marked  with  a  tack.  All 
angles  are  double  centered.  The  sight  on  the  surface  may 
of  course  be  several  hundred  feet  but  the  line  projected 
down  into  the  tunnel  cannot  be  wider  than  the  shaft,  and 
from  this  short  base  in  the  bottom  the  underground  lines 
must  be  run  with  such  a  degree  of  accuracy  that  the  hole 
at  the  starting  point  will  be  dug  in  the  proper  place.  Hav- 
ing marked  the  base  line  on  the  surface  set  the  transit 
over  one  of  the  points  at  the  edge  of  the  shaft  and  sight 
down  the  shaft,  placing  a  hub  and  tack  there.  For  this 
purpose  it  is  often  necessary  to  have  an  extra  telescope  on 
the  transit,  swinging  clear  of  the  edge  of  the  plate.  When 
the  hub  is  set  below,  take  the  transit  to  the  bottom  and  from 
a  backsight  on  the  tack  above,  project  the  line  through  the 
tunnel. 

Work,  such  as  that  just  described,  cannot  always  be 
accomplished  because  of  the  depth  of  the  shaft.  It  then 
becomes  necessary  to  set  points  in  the  bottom  of  the  shaft 
by  using  plumb-bobs.  These  bobs  should  weigh  not  less 
than  ten  pounds  and  be  suspended  by  fine  piano  wires 
instead  of  cord,  provided  with  reels  at  the  top  to  keep  the 
points  off  the  ground.  A  cord  is  stretched  across  the  top 
of  the  shaft  from  one  point  to  the  other.  The  heavy  bobs 
are  then  let  down  on  each  side  of  the  shaft,  as  close  as 
possible  to  the  sides  without  touching  same.  The  bobs 
must  be  immersed  in  buckets  of  oil  to  prevent  vibration  and 
swinging.  When  the  lines  are  perfectly  still  the  surveyor 
goes  back  into  the  tunnel  and  gets  himself  in  line  with  the 
plumb  lines,  which  are  illuminated  by  his  helpers.  His 


ENGINEERING   SURVEYING  335 

transit  is  set  up  and  leveled  and  a  sight  taken  on  the 
wires.  If  he  sees  both,  he  moves  the  transit  to  one  side  and 
tries  again.  By  trial  he  finally  gets  in  a  position  where 
one  line  only  is  seen,  the  other  being  hidden  behind  it. 
He  reverses  the  telescope  and  tries  again,  proceeding  in 
this  manner  until  he  is  satisfied  the  center  of  the  transit 
is  exactly  in  line  with  the  two  wires  and  that  the  instru- 
ment is  level.  He  then  sets  two  points,  one  ahead  and  one 
back  of  the  instrument  as  points  on  his  underground  base. 

The  student  is  referred  for  more  information  on  mining 
surveys  to  the  best  American  books  on  the  subject: 

"  Underground  Surveying,"  by  L.  W.  Trumbull,  E.  M. 
($3.00)  and  "  Mine  Surveying,"  by  Edward  B.  Durham, 
E.M.  ($3.50.) 

The  best  English  book  was  written  by  the  late  Bennett  H.    \ 
B  rough. 

HYDROGRAPHIC   SURVEYS 

Hydrographic  surveys  are  surveys  of  rivers,  lakes  and 
harbors.  The  survey  of  the  shore  line  is  made  in  the 
customary  manner  of  traverse  surveys,  the  hydrographic 
work  being  the  survey  of  the 
land  under  the  water.  Some- 
times in  making  surveys  of 
rivers  a  method  of  triangula- 
tion  may  be  used  as  shown  in 
Fig.  215. 

Assistants   first   set   Stakes  on    FlG'  2IS-     Triangulation  survey 

,        .  ,         c    , ,  , .  of  river  banks. 

each  side  ot  the  river  at  dis- 
tances approximately  equal  to  the  width  of  the  river.  The 
distance  AB  is  carefully  measured  with  a  steel  tape  and 
the  transit  is  set  on  each  stake  on  that  side  of  the  river. 
From  A,  with  a  foresight  on  B,  angles  are  taken  to  stakes  I 
and  2.  The  transit  is  then  set  on  B  and  from  a  backsight 
on  A,  angles  are  read  to  stakes  I,  2,  3  and  C.  The  transit 
is  then  set  on  C  and  from  a  backsight  on  B  angles  are  read 
to  stakes  2,  3,  4  and  D.  Proceeding  in  this  manner  for  a 
mile  or  so  another  distance  is  carefully  measured  with  a 
steel  tape.  By  plane  trigonometry  the  lengths  of  all  the 
lines  are  computed  and  the  new  base  line  is  also  computed 
as  a  check  on  the  work.  The  work  is  carried  forward  from 


336 


PRACTICAL  SURVEYING 


the  measured  length  of  the  new  base  line,  these  measured 
bases  being  introduced  at  fairly  regular  intervals  to  keep 
errors  within  reasonable  limits.  The  limit  of  error  is  first 
fixed  for  the  work  and  if  the  error  at  any  base  line  is  too 
great  it  may  be  wise  to-  have  the  measured  bases  at  closer 
intervals.  In  the  figure  EF  is  a  measured  base.  The 
distance  to  the  shore  line  is  measured  on  offset  lines  from 
the  numbered  stakes,  or  the  helpers  on  the  banks  may  have 
stadia  rods  and  the  shore  line  be  located  thus  from  the 
transit. 

Soundings  may  be  made  from  boats,  men  in  the  boats 
holding  stadia  rods  which  are  read  from  a  station  on  shore. 
This  is  only  possible  in  still  water.  For  the  proper  sound- 
ing and  survey  of  lakes  and  harbors  along  the  shores  of 
lakes  or  the  sea,  several  methods  are  in  vogue. 

In  one  the  boat  containing  the  men  making  the  soundings 
keeps  as  closely  as  possible  to  a  line  by  means  of  three  stakes 
set  on  shore  while  two  observers  at  stations  on  shore  read 
angles  from  a  base  line  to  the  boat  with  transits  or  sextants. 
Sometimes  the  boat  is  rowed  at  a  certain  rate  on  a  line 
determined  by  sighting  on  range  stakes,  and  soundings  are 
taken  at  definite  intervals  of  time.  Sometimes  an  observer 
in  the  boat  takes  angles  with  a  sextant  to  three  points  on 
shore,  thus  introducing  the  "Three  Point  Problem,"  one 
of  the  standard  problems  inserted  in  textbooks  paying  con- 
siderable attention  to  the  mathematical  side  of  surveying. 
Problem.  —  Given  three  points  in  a  triangle  and  the  dis- 
tances between  them  AB  =  317  it.,  AC  =  308  ft.,  and 
BC  =  478  ft.; also  the  angles  at  a  point 
D  which  these  distances  subtend  in  the 
same  plane  with  them,  i.e.,  ADB  =  24° 
50',  and  ADC  =  27°  44';  to  find  the 
distance. of  the  station  D  from  each  of 
them. 

Construct  the  triangle  ABC  and  on  the 
line  BC,  set  off  at   C  the  angle  BCd  = 
ADB  =  24°   50';    and  at  B  set   off  the 
FIG.    216.      Three-     angle   CBd  =  ADC  =  2?°  ^        pomt  d 

is  located  by  the  intersection  of  the  lines 
from  B  and  C.  Through  the  three  points  B,  d,  C  draw 
a  circle.  From  A  draw  a  line  through  d  and  produce  it 


ENGINEERING   SURVEYING  337 

to  an  intersection  with  the  circumference,  thus  locating  the 
point  D.  Draw  lines  from  D  to  B  and  C. 

Three  sides  being  known  in  the  triangle  ABC,  the  angle  B 
=  39°  25'  14.6" ';  then  ABd  =  ABC  +  dBC  =  67°  9'  14",  when 
^4  and  d  are  on  different  sides  of  BC,  or  =  11°  41'  14.6", 
when  ^4  and  d  are  on  the  same  side  of  BC  as  in  the  present 
case. 

In  the  triangle  BCd  are  given  the  side  BC  and  the  angles 
B  and  C.  Then  the  side  Bd  =  252.7  ft. 

In  the  triangle  ABd  are  given  the  sides  AB  and  Bd 
with  the  included  angle  ABd.  Then  the  angle  AdB  = 
I3i°  53'  53",  andBAd  =  36°  25'  53". 

Then  in  the  triangle  ABD  are  given  the  angles  and  the 
side  AB.  We  find  BD  =  448.066  ft.,  and  AD=  661.738 
ft.  In  the  triangle  DBC  are  given  the  angles  and  the 
side  BC.  We  find  DC  =  591-5^3  ft. 

If  the  triangle  ABC  is  reversed  so  the  point  A  is  the 
point  nearest  D,  the  angle  BAd  =  46°  47'  32.2";  then  BD 
=  550.153  ft.,  AD  =  282.25  ft.  and  CD  =  528.4  ft. 

1.  If  D  be  within  the  triangle,  as  at  d,  make  the  angles 
BCD  and  CBD  =  supplements  of  BaA  and  AdC. 

2.  When  D  is  in  one  of  the  sides,  describe  a  segment  on 
BC  containing  the  given  angle. 

3.  If  A  and  B  be  in  a  straight  line  with  D,  then  BC  and 
CA  subtend  the  same  angle  BDC.     Solve  the  triangle  DBC 
after  finding  the  angle  at  B. 

4.  If  the  three  points  A,  B,  C  lie  in  a  straight  line,  the 
first  operation  will  not  be  required.     The  other  operations 
are  unchanged. 

When  making  soundings  of  a  river  an  approved  method 
is  to  stretch  a  cord  across,  having  knots  at  definite  inter- 
vals and  row  a  boat  along  the  line,  taking  soundings  at 
each  knot.  This  is  possible  only  in  rivers  with  sluggish 
current.  Another  method  is  to  set  stations  along  the 
bank  at  distances  about  equal  to  the  width  of  the  river  and 
row  a  boat  across  at  an  angle  in  each  of  the  squares  thus 
formed,  taking  soundings  at  as  nearly  regular  intervals  as 
possible.  Each  square  has  two  diagonals  and  when  the 
notes  are  plotted  it  is  assumed  that  the  spaces  between 
soundings  are  equal  to  the  length  of  the  diagonal  divided 
by  the  number  of  soundings.  In  addition  to  the  diagonal 


338  PRACTICAL  SURVEYING 

lines,  soundings  are  taken  on  a  straight  line  normal  to  the 
banks  of  the  river  from  the  stake  on  one  bank  to  the  stake 
on  the  opposite  bank.  When  studying  a  navigable  river 
for  the  purpose  of  locating  a  bridge,  the  government  re- 
quires an  accurate  map  of  the  banks  for  half  a  mile  below 
and  one  mile  above  the  proposed  bridge  site.  Soundings 
must  be  shown  at  approximately  100  ft.  intervals  for  the 
entire  distance  on  the  map.  The  right-hand  bank  of  the 
river  is  the  bank  on  the  right  hand  of  the  observer  when 
traveling  with  the  current. 

Current  observations  must  also  be  given  in  hydrographic 
surveys  of  rivers.  These  are  best  obtained  by  measuring 
a  definite  base  line  along  one  bank  of  the  river  and  sight- 
ing across  the  river  perpendicularly  at  each  end.  At  the 
upper  end  floats  are  released  and  timed  as  they  cross  the 
upper  line,  being  timed  again  as  they  pass  the  lower  line. 
Bottles  partly  filled  with  water  or  sand  and  having  in  them 
small  rods  carrying  flags,  make  good  floats  for  the  purpose. 
One  bottle  with  a  white  flag  should  be  traced  down  the 
middle  of  the  river,  one  with  a  red  flag  close  to  the  right 
bank  and  one  with  a  blue  flag  close  to  the  left  bank.  To 
avoid  confusion  let  the  middle  float  go  and  after  the  recorder 
makes  his  entry  of  the  time,  release  a  second  float,  and 
when  that  record  is  made,  the  third.  The  rate  of  flow  is 
generally  given  in  feet  traveled  per  second.  To  determine 
accurately  the  amount  of  water  flowing  in  a  stream  it  will 
be  necessary  to  use  more  than  three  floats  to  get  the  stream 
lines,  and  in  each  line  of  floats  there  must  be  three:  One 
to  give  the  velocity  near  the  surface,  one  the  velocity  at 
about  half  the  depth  and  one  to  give  the  velocity  close  to 
the  bottom,  so  that  for  such  work  it  will  be  necessary  to 
first  make  soundings. 

To  obtain  the  velocity  at  different  depths  the  floats  will 
consist  of  very  thin  rods  —  wire  is  best — with  flags  at  the 
upper  ends  and  weights  at  the  bottom,  regulated  to  keep 
at  a  definite  depth,  which  is  not  a  difficult  matter.  Sound- 
ing lines  are  prepared  so  they  will  not  stretch  when  wet, 
by  wetting  and  wrapping  around  a  tree  or  post  as  tightly  as 
possible  and  leaving  there  until  dry.  This  operation  is 
repeated  several  times  and  takes  out  all  the  stretch,  after 
which  they  may  be  measured  off  in  five-foot  lengths,  with 


ENGINEERING  SURVEYING 


339 


colored  cloths  for  markers  amd  small  tags  to  indicate  the 
length.  The  weight  on  the  sounding  lines  should  be  very 
heavy  so  the  line  will  be  taut  and  there  may  be  a  reasonable 
certainty  that  the  depth  obtained  is  vertical  and  not  in- 
clined. Tags  at  five-foot  intervals  are  all  that  are  neces- 
sary, for  lesser  intervals  may  be  measured  with  a  rule  by 
the  observer  in  the  sounding  boat. 

Soundings  in  water  are  wanted  for  a  number  of  reasons 
but  an  engineer  or  surveyor  in  private  practice  usually 
does  such  work  to  ascertain  the  yardage  in  dredging  oper- 
ations. The  bottom  must  be  first  surveyed  and  after  the 
dredging  is  completed,  or  when  a  progress  estimate  is 
wanted,  must  be  again  surveyed.  A  method  used  for 
such  work  is  to  make  soundings  from  a  scow,  the  positions 
occupied  by  the  scow  being  obtained  from  the  shore. 

Fig.  217  is  a  diagrammatic 
representation  of  a  sounding 
device  used  for  such  work. 
Several  graduated  rods  are 
attached  to  the  scow  by  pass- 
ing through  holes  in  boards 
projecting  over  the  edges,  the  ^fW^r 

lower  ends  of  the  rods  being    ^ 


\ 


FIG.  217.     Sounding  from  barge. 


fastened  to  inclined  pieces. 

These    inclined    pieces    are 

hinged  at  the  upper  end  where  they  are  fastened  to  the  scow 

and  the  lower  end  of  each  drags  on  the  bottom.     Every 

rise  and  fall  in  the  bottom  is  read  by  an  observer  on  the 

scow  as  the  area  is  "swept"  over. 

For  complete  information  on  hydrographic  work  the  stu- 
dent is  referred  to  "  Hydrographic  Surveying,"  by  Samuel 
H.  Lea  ($2.00). 

ROUTE   SURVEYS 

Route  surveys  are  made  for  the  location  of  railways, 
highways,  canals,  pipe  lines,  etc.,  where  a  study  must  be 
made  of  the  possibilities  for  obtaining  a  proper  grade. 
With  railways  a  one  per  cent  grade  (i  ft.  rise  in  100  ft.  of 
distance)  is  steep.  For  common  highways  a  5  per  cent 
grade  is  about  the  limit  on  good  roads,  although  grades 
of  10  per  cent  are  not  uncommon.  The  grade  of  an  electric 


340  PRACTICAL  SURVEYING 

road  may  be  considerably  higher  than  that  of  a  steam  road, 
but  pulling  trains  and  wagons  up  a  grade  is  expensive, 
amounting  to  a  vertical  lift  in  a  given  distance  equal  to  the 
difference  in  elevation  between  the  ends.  The  grade  for 
drainage  ditches  should  be  as  great  as  possible  in  order  to 
carry  the  water  without  washing  the  banks  and  cutting 
gullies  in  the  side  of  the  hill.  This  is  true  also  of  ditches 
used  in  mining  operation.  Open  earth  ditches  have  a  very 
steep  grade  when  it  exceeds  10  ft.  in  one  mile  and  not 
all  soil  will  stand  as  much  grade.  For  drainage  ditches  it 
is  frequently  difficult  to  get  a  grade  greater  than  2  ft. 
in  one  mile.  In  irrigation  work  the  problem  is  to  take  the 
water  to  the  highest  point  on  the  tract  to  be  irrigated  and 
when  this  tract  is  reached  to  keep  it  high  to  serve  the  great- 
est possible  area.  If  the  grade  of  an  irrigation  ditch  is  too 
light  it  will  silt  up  rapidly  and  aquatic  plants  will  grow 
and  interfere  with  the  flow.  The  problem  is  then  to  nicely 
adjust  the  grade  to  keep  the  ditch,  or  canal,  on  high  ground 
and  yet  have  a  rapid  enough  flow  to  prevent  undue  silting 
and  growth  of  grass.  Three  feet  in  one  mile  is  a  big  grade 
for  an  irrigation  ditch  and  some  have  a  fall  of  only  8  ins. 
to  the  mile. 

A  simple  method  of  survey  for  roads,  ditches  and  canals 
consists  in  setting  grade  pegs  according  to  methods  given 
in  the  chapter  on  leveling.  The  line  follows  very  closely 
the  contours  of  the  surface  and  is  necessarily  crooked. 
A  better  way  is  to  set  the  grade  pegs  farther  apart,  100  to 
200  ft.,  and  follow  this  line  with  a  transit  line  regularly 
stationed.  Elevations  of  the  ground  at  each  station  are 
taken  and  also  side  slopes  so  contours  may  be  placed  on 
the  map.  A  grade  line  is  then  selected  to  give  as  long 
straight  sections  and  as  easy  curves  as  may  be  advisable. 
The  grade  line  is  plotted  on  profile  paper  and  the  cuts  and 
fills  balanced.  The  line  is  then  staked  out  on  the  ground 
from  the  field  notes  picked  out  on  the  map. 

A  railway  survey  proceeds  upon  practically  the  same 
system  but  the  locating  engineer  goes  ahead  and  picks 
out  the  route  by  eye  without  running  preliminary  levels, 
using  a  hand  level  to  keep  him  close  to  grade.  On  this 
line  selected  by  the  chief  of  party  the  transitman  directs 
the  chainmen  in  setting  stakes  at  stations  one  hundred 


ENGINEERING   SURVEYING 


341 


common 


feet  long  and  takes  angles  at  all  changes  in  direction.  The 
leveler  follows  and  gets  the  elevations  at  each  stake.  The 
topographer  —  slopeman  he  is  called  on  some  roads  —  takes 
the  side  slopes  and  the  draftsman  plats  all  the  information 
on  a  map,  the  leveler  making  a  profile  each  evening  of  the 
day's  work.  On  the  map  the  chief  of  party  picks  out  the 
line,  which  is  then  run  in  and  "located"  on  the  ground. 

For  full  information  on  field  and  office  methods  of  mak- 
ing route  surveys  the  student  is  referred  to  "Railroad 
Location  Surveys  and  Estimates,"  by  F.  Lavis,  M.  Am. 
Soc.  C.  E.  ($3.00). 

CONTOURS 

Many  surveys  are  made  for  the  purpose  of  obtaining 
information  about  the  shape  of  the  surface  of  the  ground. 
This  is  called  topographical  surveying.  The 
method  today  for  showing 
the  surface  of  the  ground  is 
by  means  of  contours  and 
every  surveyor  should  know 
how  to  make  surveys  for  the 
purpose  of  preparing  con- 
tour maps  and  should  know 
how  to  use  such  maps.  A 
surveyor  given  a  tract  of 
land  to  subdivide  should 
first  prepare  a  contour  map 
in  order  that  he  may  lay  the 
roads  on  proper  grades  and 
take  care  of  sewerage  and 
drainage.  Properly  made 
contour  maps  may  be  used 
also  in  earthwork  calcula- 
tions. 


FIG.  218.     Contours  and  profile  of  hill. 


In  the  upper  part  of  Fig.  218  is  shown  a  contour  map  of 
a  hill  and  the  lower  part  shows  a  side  view  of  the  hill  with 
the  contour  lines  dotted  across  the  face.  Contours  are 
level  lines  defining  the  shape  of  the  surface  of  a  hill,  or 
hollow.  Assume  that  the  hill  in  Fig.  218  was  submerged 
by  water  which  stood  at  a  depth  of  26  ft.  long  enough 
to  leave  a  mark  on  the  hillside  at  that  height.  The  water 


342 


PRACTICAL   SURVEYING 


then  receded  to  a  depth  of  25  ft.  and  left  a  mark.  It 
receded  5  ft.  at  a  time,  standing  at  each  level  long 
enough  for  debris  to  collect  on  the  hillside  at  each  step, 
so  that  the  respective  elevations  were  marked  on  the  sur- 
face. Water  surfaces  are  always  level  so  each  line  of  debris 
marked  a  level  line  around  the  hill  at  the  elevations  shown. 
A  side  view  shows  the  lines  as  level,  while  the  plan,  or 
map,  gives  the  shape  of  the  hillside. 

In  the  upper  part  of  Fig.  219  is  shown  a  map  of  a  hill- 
side, having  two  projections.     The  straight  lines  show  lines 


FIG.  219.     Selecting  grades  on  contours. 

connecting  points  A  and  B  and  C  and  D.  The  contours 
are  marked  on  the  map  and  it  is  desired  to  obtain  a  uni- 
form grade  connecting  C  and  D.  First  draw  a  straight 
line  from  C  to  D  and  on  this  line  note  the  elevations  of  the 
points  /  and  P.  On  the  profile  paper  below  plot  these 
points,  and  also  the  elevations  of  intermediate  points  and 
the  wavy  line  C,  /,  P,  D  is  obtained.  A  straight  line 
therefore  will  not  give  a  uniform  grade.  Following  the 
dotted  line  and  plotting  it  on  the  profile  a  uniform  grade 
is  obtained  so  that  it  is  represented  by  the  straight  line  CD 
on  the  profile. 

The  dotted  line  is  obtained  in  actual  practice  by  drawing 


ENGINEERING  SURVEYING  343 

the  straight  line  on  the  profile  and  then  projecting  the 
points  where  it  intersects  the  horizontal  lines  on  the  profile 
paper  up  to  an  intersection  with  the  contour  lines  on  the 
map.  The  intersection  being  obtained  with  the  contour  line 
in  each  case,  a  dotted  line  is  drawn  from  one  contour  to 
the  next  as  shown. 

The  line  AB  is  similarly  studied.  First  the  straight  line 
was  drawn  on  the  map  resulting  in  the  very  wavy  line  A, 
L,  N,  M,  B.  Next  a  crooked  line  on  the  map  was  selected 
resulting  in  the  wavy  profile  A,  E,  F,  H,  B,  which  has 
smaller  depressions  and  less  marked  elevations  than  the 
first  line.  To  obtain  a  proper  grade  line  on  the  map  a 
straight  line  should  be  drawn  on  the  profile  from  A  to  B 
and  this  line  projected  up  to  the  map. 

In  making  route  surveys  all  data  are  obtained  which  are 
necessary  to  show  the  contours  for  some  distance  either 
side  of  the  surveyed  line.  A  grade  is  picked  out  on  the 
profile  made  by  the  leveler  and  the  point  where  the  grade 
crosses  each  contour  is  marked  by  stations.  These  points 
are  laid  off  on  the  contour  map  and  a  line  sketched  in  to 
indicate  the  line  of  uniform  grade.  The  chief  engineer,  or 
chief  of  party,  then  lays  a  line  as  close  to  this  as  possible 
so  cuts  and  fills  will  be  fairly  equalized  and  the  line  be  as 
free  as  possible  from  great  curvature  and  abrupt  changes 
in  direction. 

Fig.  220  is  reproduced  from  an  article  by  Dr.  W.  G.  Ray- 
mond, in  Vol.  VI,  Transactions  of  the  Technical  Society  of 
the  Pacific  Coast,  p.  72  (1889).  The  description  is  as  fol- 
lows: "A  small  reservoir  is  to  be  built  on  a  hillside,  and  will 
be  partly  in  excavation  and  partly  in  embankment.  The 
contours  are  spaced  5  ft.  apart.  The  top  of  the  wall  — 
shown  by  the  full  lines  making  the  square  —  is  10  ft.  wide 
and  at  an  elevation  of  660  ft.  The  reservoir  is  20  ft.  deep, 
with  inside  and  outside  side  slopes  of  2  to  I,  making  the 
bottom  elevation  640  ft.  and  20  ft.  square,  the  top  being 
100  ft.  square  on  the  inside.  The  dotted  lines  are  contours 
of  the  finished  surfaces  inside  and  outside  the  reservoir. 
The  areas  in  fill  all  fall  within  the  broken  line  marked 
abcdefghik,  and  the  cut  areas  all  fall  within  the  broken  line 
marked  abcdejgo.  These  broken  lines  are  grade  lines. 
The  areas  of  fill  and  cut  are  readily  traced  by  following 


344 


PRACTICAL  SURVEYING 


the  closed  figures  formed  by  contours  of  equal  elevation, 
thus: 

At  640  ft.  level  the  area  in  fill  is  pst. 

At  650  ft.  level  the  area  in  fill  is  Imnuvxl. 

At  650  ft.  level  the  area  in  cut  is  I,  2,  3  uxl. 

The  other  areas  are  as  easily  traced.     The  closed  figures 
formed  by  the  intersecting  contours  of  equal  elevation  are 


640 


640 


(680 


08Q  070  6(50  650. 

FIG.  220.     Computing  earthwork  from  contour  map. 

horizontal  areas  of  cut  or  fill  separated  by  the  common 
vertical  and  perpendicular  distances  between  successive 
contours.  These  areas  may  be  measured  and  the  quanti- 
ties intercepted  between  computed  by  the  prismoidal,  or 
other  formula,  used  in  earthwork  computations.  Where 
the  grade  contours  do  not  intersect  natural  contours  of 
equal  elevation,  but  themselves  form  closed  areas,  those 
areas  are  to  be  measured. 

The  author  worked  for  men  who  actually  ran  out  the 
contours  on  the  ground  for  the  purposes  of  map  making. 
A  level  was  used  to  insure  the  lines  being  kept  at  the  proper 
elevation  and  stakes  were  set  on  these  level  lines  at  regular 


ENGINEERING  SURVEYING 


345 


intervals.  A  transit  party  followed  the  level  party  taking 
angles  and  making  measurements  to  stakes  so  the  lines 
could  be  plotted  on  paper.  This  is  a  slow  method  and 
seldom  used. 

A  method  often  followed  is  to  set  stakes  in  rectangles  or 
squares  as  already  described,  taking  the  elevation  of  the 
ground  at  the  stake.  This  information  is  platted  in  the 
office  and  contours  drawn.  On  extensive  surveys  the  rec- 
tangles may  have  sides  several  hundred  feet  in  length, 
while  for  surveys  of  small  tracts,  such  as  city  lots,  small 
reservoirs,  etc.,  the  dimensions  may  be  from  10  to  50  ft. 
It  is  not  necessary,  however,  to  set  many  stakes  in  making 
topographical  surveys. 

PLANE  TABLE  WORK 

One  of  the  oldest  map-making  methods  is  that  of  the 
plane  table.  It  is  in  high  favor  for  certain  classes  of  work 
today  and  a  number  of  government  engineers  and  sur- 
veyors and  men  with  government  experience  display  great 
sensitiveness  if  plane 
tabling  is  criticized. 

The  author  at  one 
time  was  a  fairly  good 
"plane  tabler"  but  must 
confess  a  strong  prefer- 
ence for  the  transit  and 
stadia  for  making  topo- 
graphical surveys.  The 
transit  is  a  universal  sur- 
veying instrument,  while 
the  plane  table  costs  as 
much  and  is  good  for  one 

kind  of  work  only,  work  FlG.  22I     Plane  table 

that  does  not  come  often 

enough  to  the  average  engineer,  or  surveyor,  to  justify  the 
expenditure  provided  he  has  a  transit  equipped  with  stadia 
wires.  The  author  has  never  been  able  to  quite  see  where 
the  plane  table  possesses  any  advantage  over  the  transit  and 
stadia,  especially  since  modern  plane  tables  have  telescopes 
equipped  with  stadia  wires,  and  depend  less  than  formerly 
upon  the  principle  of  intersection. 


346  PRACTICAL  SURVEYING 

The  plane  table  is  illustrated  in  Fig.  221.  A  sheet  of 
paper  is  fastened  to  the  board  and  the  map  drawn  as 
rapidly  as  the  sights  are  taken.  Damp  weather  and  very 
dry,  hot,  sunshiny  weather  affects  the  paper  and  the  working 
hours  during  the  day  are  not  long.  The  map  is  accurate 
only  for  the  scale  to  which  it  is  drawn.  A  reduction  of 
course  increases  the  apparent,  but  not  the  relative,  accu- 
racy while  all  errors  are  multiplied  by  an  enlargement  of  the 
map. 

When  a  topographical  survey  is  made  with  transit  and 
stadia,  notes  are  placed  in  a  book  with  such  sketches  as  are 
advisable  and  from  these  notes  maps  may  be  made  to  any 
scale.  The  degree  of  accuracy  is  independent  of  the  scale 
and  depends  on  the  care  with  which  the  notes  are  taken  and 
recorded.  The  work  may  be  done  in  any  weather  when 
the  graduations  can  be  read  on  the  rods  and  for  more 
hours  in  the  day  than  are  possible  with  a  plane  table.  For 
most  of  the  work  of  this  nature  transit  work  is  far  more 
rapid  than  plane  table  work. 

For  complete  information  on  plane  table  work,  as  well  as 
on  all  kinds  of  topographical  surveys  the  student  is  referred 
to  "Topographic,  Trigonometric  and  Geodetic  Surveying," 
by  Herbert  M.  Wilson,  M.  Am.  Soc.  C.  E.  ($3.50). 

STADIA  WORK 

The  writer  assumes  that  every  surveyor  has  a  transit 
equipped  with  stadia  wires,  so  a  description  of  methods  is 
in  place  in  this  book.  The  principles  underlying  the  use 
of  stadia  wires  were  explained  in  a  preceding  chapter. 

To  make  a  stadia  survey  for  topographical  purposes,  a 
number  of  points  are  selected  at  which  the  transit  is  set 
and  from  each  point  side  shots  are  taken  to  locate  the  in- 
equalities of  the  ground,  natural  objects,  fences,  buildings, 
etc.  The  first  step  is  the  selection  of  the  instrument  sta- 
tions. The  maximum  length  of  sight  from  station  to 
station  should  not  exceed  1000  ft.,  and  the  best  average 
length  for  sights  is  less  than  700  ft.  The  instrument  sta- 
tions therefore  should  be  selected  with  these  limits  in  mind 
and  each  should  be  on  a  point  from  which  as  many  side 
shots  as  possible  may  be  taken.  After  the  instrument  sta- 


ENGINEERING  SURVEYING  347 

tions  are  selected  it  is  a  good  plan  to  run  a  line  of  levels 
with  a  good  level  and  obtain  the  ground  elevation  at  each 
point.  This  then  leaves  the  angular  elevations  as  merely 
a  check.  If  time  is  an  object  and  the  survey  is  not  very 
important  the  leveling  may  be  omitted  and  the  elevations 
obtained  by  means  of  vertical  angles. 

After  each  point  is  selected  and  a  stake  set,  which  does 
not  require  a  tack  in  the  top,  the  transit  is  set  up  on  each 
station  precisely  as  though  a  closed  farm  survey  was  being 
made.  The  distance  is  read  forward  and  back  on  each 
course,  also  the  vertical  angles.  The  horizontal  angles 
are  usually  azimuths  and  when  this  survey  is  completed  the 
survey  is  closed  in  the  usual  way  and  the  errors  distributed. 
Then  each  point  is  given  its  co-ordinate  numbers  so  it  may 
be  plotted  by  latitudes  and  departures.  When  the  frame- 
work is  platted  the  side  shots  from  each  station  are  platted. 
In  addition  to  the  readings  taken  with  the  stadia  rods 
intersection  angles  are  read  from  each  station  to  as  many 
other  stations  as  can  be  seen,  simply  to  check  the  work. 

From  the  foregoing  paragraph  the  student  must  not 
imagine  that  the  work  is  gone  over  twice;  once  for  the 
framework  and  once  for  the  side  shots.  When  each  sta- 
tion is  occupied  and  the  instrument  oriented  all  the  sights 
possible  from  that  station  are  read  and  recorded.  When 
the  office  work  is  done  the  reductions  for  distance  and  eleva- 
tion are  made  first  for  the  station  framework  and  the  results 
plotted  before  the  work  of  reducing  and  plotting  the  side 
shots  begins. 

The  instrument  is  first  oriented.  This  means  the  line 
through  o°  and  180°  lies  in  the  selected  meridian,  which 
may  or  may  not  be  the  true  meridian.  It  is  convenient, 
however,  to  use  the  true  meridian  so  angles  may  be  checked 
by  the  needle  and  to  assist  the  man  who  makes  the  map  to 
keep  himself  located.  People  being  trained  from  infancy 
to  look  to  the  north  as  a  starting  point,  the  average  man  is 
usually  unable  to  read  a  map  until  it  is  oriented,  a  habit  the 
surveyor  loses  with  experience.  The  instrument  properly 
oriented,  a  sight  is  taken  to  the  station  ahead,  the  azimuth 
read,  the  rod  read  and  the  vertical  angle  read,  all  being 
recorded  in  the  field  book.  Keeping  the  lower  clamp  tight 
the  instrument  is  swung  to  right  and  left  and  the  rod  read 


348  PRACTICAL  SURVEYING 

as  it  is  held  on  different  points  by  the  rodman,  or  men,  for 
on  some  ground  an  instrument  man  can  keep  two  rodmen 
busy.  When  he  has  three  or  four  rodmen  busy  he  must 
have  a  recorder  at  his  side  to  whom  he  can  call  off  the  read- 
ings on  the  rod  and  the  angles.  When  all  the  sights  are 
taken  which  are  thought  necessary  from  this  station  the 
plates  are  set  to  the  angle  first  read  to  the  station  ahead 
and  the  telescope  pointed  properly.  It  will  be  found  fre- 
quently that  the  instrument  has  been  slightly  disturbed  by 
the  intermediate  sighting  so  this  must  be  corrected  by  the 
lower  tangent  screw,  thus  not  disturbing  the  angle.  How- 
ever, if  the  plate  is  set  to  read  the  correct  azimuth  it  will 
not  be  necessary  to  touch  the  lower  tangent  screw,  for  the 
instrument  is  then  carried  ahead  to  the  next  station. 

Set  the  instrument  over  the  next  station.  Level  care- 
fully and  reversing  the  telescope  sight  back  on  the  station 
just  left.  Without  in  any  way  disturbing  the  angle  on  the 
plate  get  an  exact  sight  by  means  of  the  lower  clamp  and 
tangent  screws.  Having  clamped  the  transit  in  position, 
read  the  angle,  and  if  any  change  has  taken  place,  bring 
the  plate  and  vernier  to  indicate  the  correct  reading  and 
then  re-set  the  line  of  sight  on  the  last  station.  When 
this  is  done  and  the  telescope  transited  to  read  in  a  for- 
ward direction  it  will  lie  in  the  line  of  sight  between  the 
two  stations  with  the  proper  azimuth  indicated  on  the 
plates.  Now  unclamp  the  plate,  read  the  azimuth,  vertical 
angle  and  rod  to  the  next  station,  record  the  readings,  and 
take  the  side  shots  as  was  done  from  the  previous  station. 
Some  writers  do  not  advise  the  reversing  of  the  telescope, 
thinking  it  best  to  set  the  plate  on  vernier  B  to  read  an 
angle  180°  different  from  the  forward  azimuth  from  the 
previous  station.  With  this  angle  set  on  the  plate  the  back- 
ward sight  is  taken  on  the  previous  station  with  the  tele- 
scope erect,  after  which  the  plate  clamp  is  loosened  and 
the  instrument  may  be  revolved  on  its  vertical  axis  to  take 
sights  in  all  directions.  The  author  could  never  see  a  good 
reason  for  this.  It  increases  the  number  of  operations; 
it  renders  mistakes  possible  because  the  instrument  man 
might  get  mixed  up  in  adding  the  180°,  and  it  causes  loss 
of  time.  To  keep  the  azimuth  set  on  the  plate  and  reverse 
the  telescope  introduces  an  automatic  check. 


ENGINEERING  SURVEYING  349 

All  the  line  readings  should  be  taken  forward  and  back 
and  all  the  elevations  checked  by  the  positive  and  nega- 
tive vertical  angles.  The  side  shots  do  not  require  such 
repeating  and  the  instrument  man  must  be  constantly  on 
guard  to  avoid  setting  down  wrong  readings.  The  side 
shots  are  taken  to  every  object  it  is  desired  to  locate  on  the 
map,  fence  corners,  corners  of  buildings,  bridges,  etc.,  and 
it  is  well  to  make  a  sketch  of  these  on  the  opposite  page. 
The  principal  side  shots,  however,  are  taken  to  inequalities 
of  the  surface  of  the  ground  to  obtain  their  location  and 
elevation.  An  experienced  instrument  man  with  expe- 
rienced rodmen  will  find  300  shots  a  day  about  an  average 
day's  work,  except  in  such  rare  cases  where  more  than  50 
shots  can  be  taken  from  one  station.  The  average  number 
of  sights  from  one  station  in  average  farming  country  with 
the  average  number  of  natural  objects  is  about  20  to  a 
station,  the  number  from  some  stations  being  less  than  5. 
A  fault  with  beginners  is  to  take  too  many  sights  to  obtain 
ground  elevations.  This  is  better,  however,  than  to  take 
so  few  that  the  map  has  to  be  pretty  thoroughly  "guessed 
out"  when  the  contours  are  platted.  The  rule  is  to  have 
the  rod  held  at  each  point  where  there  is  a  decided  change 
in  slope.  An  experienced  rodman  who  has  also  helped  to 
make  maps  can  cut  down  the  working  time  of  a  party 
one-half.  Plentiful  use  of  sketches  is  advisable. 

The  author  for  many  years  used  loose  sheets  in  his  field 
book  having  two  lines  ruled  at  right  angles  and  crossing 
in  the  middle  of  the  sheet.  From  this  as  a  center  ten 
circles  were  drawn  with  intervals  between  the  lines  of  one- 
tenth  of  an  inch.  Using  the  vertical  line  as  the  meridian 
when  a  sight  was  taken  to  any  fence,  building,  etc.,  and 
the  rod  read,  a  ruler  was  laid  from  the  center  of  the  circle 
at  as  nearly  as  possible  the  same  angle  with  the  meridian 
and  a  point  made  at  the  distance  indicated  by  the  rod 
reading,  the  interval  between  two  circles  being  assumed 
as  100  ft.  After  all  these  sights  to  objects  were  marked  a 
sketch  was  made  showing  each  in  its  relative  position.  The 
same  method  was  pursued  when  getting  the  edges  of  steep 
banks,  large  boulders,  etc.  Sheets,  such  as  the  author 
used,  can  be  drawn  on  tracing  cloth  by  any  surveyor  and 
printed  by  the  direct  process,  showing  dark  lines  on  a 


350  PRACTICAL  SURVEYING 

white  ground,  or  they  may  be  duplicated  by  the  mimeo- 
graph, the  hektograph  or  the  clay  process.  The  writer 
had  a  zinc  etching  made  showing  three  sets  of  circles, 
somewhat  smaller  than  recommended  above,  on  one  page 
with  heavy  division  lines,  so  sights  from  one  station  would 
not  be  confused  with  sights  from  the  other.  From  this 
cut  he  had  several  thousand  sheets  printed.  The  sheets 
being  held  in  the  fold  of  the  book  by  a  rubber  band  each 
could  be  removed  as  fast  as  it  was  filled  and  the  center  of 
the  circle  used  at  each  station  was  numbered  with  the  num- 


FIG.  222.     Stadia  survey  note  book. 

ber  of  the  station.  A  copyrighted  field  book  on  a  similar 
plan  is  sold  by  instrument  dealers,  but  as  the  author  and 
several  of  his  friends  used  the  method  for  a  number  of  years 
before  the  copyright  was  granted  for  the  book  mentioned, 
no  surveyor  need  fear  to  make  his  own  sheets.  A  variation 
on  this  was  used  by  one  surveyor,  who  had  a  rubber  stamp 
made  with  the  concentric  circles  and  the  intersecting  nor- 
mal lines.  This  stamp  was  used  on  each  page  where  it  was 
desired  to  make  sketches,  as  it  is  not  necessary  to  make 
sketches  from  each  station.  The  stamp  was  in  a  thin  metal 
case  containing  an  ink  pad  and  had  a  folded  handle  on 


ENGINEERING  SURVEYING  351 

top,  so  it  could  be  carried  in  the  vest  pocket.  A  blotter 
was  necessary  to  prevent  smudging  of  notes. 

For  ordinary  work  a  Philadelphia  leveling  rod  with  alter- 
nate hundredths  black  may  be  used.  For  long  sights 
graduations  of  a  more  pronounced  character  should  be 
used.  The  form  of  mark  on  the  face  of  a  rod  may  be 
pointed  or  square,  opinion  today  tending  towards  the  dis- 
carding of  angular  points  and  favoring  square  block-like 
graduations.  The  author  favored  a  combination  of  acute 
and  square  marks  a  few  years  ago,  but  now  prefers  the 
square  marks. 

The  graduation  shown  in  Fig.  223  was  devised  by  Prof. 
W.  H.  Burger  of  Northwestern  University.  The  gradua- 
tions for  even  feet  are  placed  on  the  left  and  those  for  odd 


FIG.  223.    Stadia  rod. 

numbered  feet  on  the  right.  Opposite  each  half  foot  is 
placed  a  dot.  *As  with  all  stadia  rods  it  is  optional  with 
the  user  whether  the  feet  are  indicated  by  figures,  it  being 
an  advantage  to  omit  them,  for  it  will  then  make  no  differ- 
ence if  the  rodman  accidentally  holds  the  rod  upside  down. 
The  instrument  man  can  readily  direct  the  middle  hair  to 
the  height  of  the  telescope  and  read  the  number  of  gradu- 
ations intercepted  between  the  stadia  wires.  Some  men 
prefer  to  have  foot  marks  on  the  rods  and  in  such  event 
they  are  one-tenth  of  a  foot  high,  the  bottom  touching  the 
division  line,  the  figure  being  placed  in  the  white  space. 

To  make  a  rod  use  a  piece  of  clear  white  pine,  well 
seasoned,  three  inches  wide  by  one  or  if  ins.  thick,  sur- 
faced and  sandpapered.  Give  it  a  thin  filler  coat  and' on 
top  of  that  three  coats  of  pure  white  paint.  Let  each  coat 
dry  thoroughly  and  sandpaper  it  before  applying  the  next 
coat.  With  a  steel  tape  carefully  measure  the  rod  and 
make  cuts  at  the  foot  divisions.  On  the  ends  of  the  rod 
place  iron  or  brass  strips  to  prevent  undue  wear  and  abra- 
sion. Make  the  pattern  for  one  foot  in  length  on  tough 
drawing  paper  and  cut  it  out  with  a  sharp  knife.  Lay 


352  PRACTICAL  SURVEYING 

this  stencil  on  the  foot  division  marks  and  with  a  fine- 
pointed  pencil  trace  the  outline  of  the  figure  on  the  painted 
surface.  Use  good  black  oil  paint  and  a  fine  camel's  hair 
brush  and  fill  in  the  marks.  A  steady  hand  is  required. 
On  the  back  of  the  rod  may  be  hung  some  device  to  insure 
its  verticality  or  the  rodman  may  carry  a  rod  level,  this 
being  as  good  a  device  as  can  be  used.  In  stadia  work  the 
judgment  of  the  rodman  cannot  be  of  service,  for  the  rod 
must  be  as  nearly  vertical  as  it  is  possible  to  get  it  and  it 
cannot  be  waved  as  in  leveling  operations.  The  length 


FIG.  224.     Protractor  for  platting  stadia  survey. 

of  the  rod  depends  upon  the  work  to  be  done,  the  writer 
having  8-ft.  and  12-ft.  rods.  Long  rods  may  be  made  con- 
venient for  carrying  by  hinging  them.  Good  stadia  rods 
may  be  purchased. 

The  notes  are  reduced  in  the  office  very  rapidly,  it  being 
best  to  use  two  men,  one  to  call  off  and  the  other  to  look 
up  the  tables,  or  use  the  diagram,  and  reduce  the  rod 
bearing  to  horizontal  distance  and  obtain  the  difference  in 
elevation.  The  difference  in  elevation  is  added  to,  or  sub- 
tracted from  (according  to  the  positive  or  negative  sign 


ENGINEERING   SURVEYING  353 

of  the  vertical  angle),  the  elevation  of  the  ground  at  the 
instrument  station.  When  the  instrument  is  set  up  the 
surveyor  measures  with  a  tape  or  rod  the  height  of  the  axis 
of  the  telescope  above  the  ground  and  directs  the  middle 
wire  to  this  height  on  the  rod.  The  line  of  sight  is  then 
parallel  to  a  straight  line  drawn  from  the  hub  to  the  foot 
of  the  rod.  In  the  office  the  last  two  columns  in  the  field 
book  are  filled. 


PLATTING   STADIA  WORK 

To  plat  the  work,  the  instrument  stations  are  first  placed 
on  paper  by  any  approved  method,  the  author  doing  this 
work  by  computing  the  latitude  and  departure  of  each  point 
and  platting  by  co-ordinates.  Through  each  station  draw 
a  fine  line  for  the  meridian  and  a  perpendicular  through  it. 

Take  a  14-in.  paper  protractor,  untrimmed,  and  cut  out 
the  interior  with  a  radius  of  5^  ins.,  making  the  diameter 
of  the  cut-out  portion  II  ins.  The  figured  graduations  go 
from  o°  to  360°,  clockwise  as  on  the  transit  plate.  Draw 
a  fine  ink  line  from  o°  to  180°  and  from  90°  to  270°,  the 
lines  terminating  at  the  edges  of  the  interior  cut-out  space. 
When  the  protractor  is  placed  on  the  perpendicular  lines 
through  a  station  so  the  lines  coincide  with  those  above 
mentioned,  the  station  is  at  the  center  of  the  graduated 
circle.  The  protractor  is  to 'be  oriented  so  angles  read  on 
it  will  coincide  with  the  same  angles  read  on  the  transit. 
Fasten  the  protractor  in  place  with  two  thumb  tacks  near 
the  upper  edge,  so  it  may  be  flopped  over  without  disturb- 
ing the  adjustment,  when  plotting  long  sights. 

Having  decided  upon  the  scale,  glue  to  the  bottom  of 
the  scale  at  the  zero  end  projecting  beyond  the  edge  from 
|  in.  to  J  in.  a  piece  of  tough  paper  J  in.  wide.  Put  a  fine 
needle  through  this  projecting  strip  at  the  zero  graduation 
and  push  the  point  through  the  station.  The  scale  can 
now  be  swung  around  the  circle  freely. 

An  assistant  calls  off  the  azimuth  from  the  field  book  and 
the  draftsman  swings  the  scale  to  this  reading  on  the  grad- 
uated circle.  The  assistant  calls  off  the  horizontal  dis- 
tance, the  draftsman  scaling  it,  and  putting  a  dot  on  the 
paper.  The  draftsman  then  writes  down  the  elevation 


354 


PRACTICAL  SURVEYING 


read  to  him  by  the  assistant.  The  shots  are  plotted  in  this 
manner  at  each  station.  Sketches  are  transferred  from 
the  field  book  to  the  map  and  the  protractor  moved  to  the 
following  station.  When  the  plotting  is  completed  each 
point  occupied  by  a  rod  shows  on  the  map  with  the  ground 
elevation,  and  all  buildings,  fences,  etc.,  are  shown  in  light 
lines.  The  next  step  is  to  put  in  the  contours. 


570 


INTERPOLATING   CONTOURS 

In  Fig.  225  is  shown  a  method  described  by  the  author 
in  Engineering  News,  May  10,  1900,  while  Fig.  226  illus- 
trates a  method  described 
by  H.  F.  Bascom,  C.E., 
in  Engineering  News,  June 
21,  1900. 

In  Fig.  225  two  points 
are  shown  with  the  eleva- 
tions. A  fine  line  is  drawn 
connecting  them.  A  piece 
of  graduated  paper  is 
placed  at  an  angle  with 
the  line  with  the  gradua- 
tions corresponding  to  the 
lower  elevation  coinciding 


FIG.  225.     Interpolating  contours, 
other  point  and 


touch    the 

squared  paper  correspond- 
ing to  the  elevation  of  the 
point.  A  straight-edge  is 
now  placed  under  the  tri- 
angle. The  triangle  is 
moved  along  the  straight- 
edge and  as  it  succes- 
sively reaches  the  num- 
bered graduations  a  point 
is  marked  at  the  inter- 
section of  the  edge  of 
the  triangle  with  the  line 


with  it.    A  triangle  is  then 
placed  so   one   edge   will 
also   the   graduation  on    the 


FIG.  226.     Interpolating  contours. 


connecting  the  plotted  points. 
In  Fig.  226  the  triangle  and  straight-edge  are  not  used. 


ENGINEERING   SURVEYING  355 

A  strip  of  ruled  paper  is  placed  with  the  edge  touching  the 
two  plotted  points.  A  second  strip  with  graduations  is 
placed  over  it  as  shown.  From  each  figured  graduation 
the  intersecting  line  on  the  under  piece  of  paper  is  followed 
to  the  edge  and  the  contour  point  marked.  Contours  are 
seldom  drawn  at  closer  intervals  than  5  ft.  in  height,  de- 
pending on  the  slope.  On  easy  slopes  the  interval  may  be 
less  than  5  ft.,  while  on  very  steep  slopes  the  interval  may 
be  as  great  as  25,  or  even  50,  ft.  When  all  the  contour 
points  are  marked  the  draftsman  draws  in  the  contours  in 
lead  pencil.  The  instrument  man  and  his  assistants,  hav- 
ing a  good  knowledge  of  the  shape  of  the  surface  of  the 
land,  look  over  the  map  and  suggest  corrections  before 
the  contours  are  inked  in.  Contours  are  usually  drawn  in 
light  brown  or  red  lines. 

PHOTOGRAPHIC   TOPOGRAPHY 

The  author,  on  May  5,  1893,  read  before  the  Technical 
Society  of  the  Pacific  Coast,  a  paper  with  the  above  title. 
The  paper  is  here  given  with  the  final  paragraph  only 
omitted.  In  the  intervening  22  years  the  camera  has  been 
used  on  many  surveys  and  its  use  is  increasing.  The 
principles,  however,  are  the  same  as  when  the  paper  was 
prepared.  The  best  book  on  the  subject  is  "Phototop- 
ographic  Methods  and  Instruments,"  by  J.  A.  Flemer,  a 
recent  work  ($5.00). 

The  object  of  this  paper  is  to  present  a  method  of  sur- 
veying which  will  be  a  valuable  auxiliary  to  plane-table 
and  stadia  work,  and  in  some  cases  is  extremely  useful  alone. 
It  is  by  no  means  new,  but  is  not  very  well  known,  and  its 
advantages  are  so  great  that  engineers  who  have  much 
topographic  surveying  to  do  should  understand  it. 

The  principle  depends  upon  the  art.  of  projecting  per- 
spective views  upon  a  horizontal  plane,  and  was  first  used 
by  French  naval  officers  in  the  beginning  of  this  century  in 
the  survey  of  coast  lines.  Perspective  drawings  were 
made  of  certain  places  from  two  or  more  positions,  and  sex- 
tant angles  taken  to  several  objects  in  the  landscape,  and 
the  angles  recorded  on  the  drawings.  These  drawings 
were  afterwards  used  in  the  mapping  of  the  shore  line,  and 


356  PRACTICAL  SURVEYING 

the  accuracy  of  the  work  depended  upon  the  skill  with 
which  the  sketches  were  made. 

The  invention  of  photography  made  it  possible  to  do 
better  work,  and  Colonel  Laussedat  of  the  French  army 
made  experiments  covering  a  period  of  years  to  perfect 
the  system.  He  published  a  work  on  the  subject  in  1865, 
and  no  improvements  have  been  made  since,  unless  multi- 
tudinous patents  for  cameras  can  be  styled  improvements. 

The  system  is  extensively  used  in  Europe,  and  is  very 
little  known  in  this  country,  where,  of  all  places,  one  would 
think  it  most  valuable  from  an  economical  standpoint. 

Instances  can  be  given  of  the  time  in  which  various  sur- 
veys have  been  made  by  this  method,  but  such  records  are 
of  no  use  unless  the  ground  were  known,  but  the  French 
engineers  generally  considered  that  a  topographical  survey 
and  map,  when  a  camera  with  horizontal  and  vertical 
circles  is  used,  can  be  made  in  one- third  the  time  required 
by  other  methods. 

Any  camera  may  be  used  provided  it  is  perfectly  level 
when  the  view  is  taken,  and  the  smallest  size  adapted  for 
the  work  is  one  with  a  5  by  8-in.  plate.  Smaller  cameras 
may  be  used  in  the  same  manner  as  a  sketching  pad,  to 
carry  away  unimportant  details,  but  are  of  little  practical 
use  unless  most  excellently  made.  All  lenses  must  be  good. 
Although  an  ordinary  camera  may  be  used,  still  it  is  better 
to  have  one  for  the  purpose,  provided  with  two  levels  on 
top  at  right  angles,  and  four  leveling  screws  beneath.  The 
box  should  be  solid,  and  focusing  done  by  means  of  the  objec- 
tive slide.  If  the  camera  has  a  compass,  or  a  horizontal 
limb,  and  a  vertical  limb,  so  that  up  or  down  sights  may  be 
taken,  it  will  be  complete. 

Glass  negatives  are  the  most  accurate  to  use,  but  films 
on  account  of  portability  are  more  convenient.  The  weight 
of  the  glass  in  the  field  is  a  drawback. 

There  are  several  adjustments  of  the  camera  which  must 
not  be  neglected.  The  first  is  called  "  the  test  for  register. " 
The  film  on  the  sensitive  plate  must  exactly  re-place  the 
surface  of  the  ground  glass.  To  do  this  set  the  instrument 
up  and  focus  for  a  distant  view.  Make  a  scratch  to  show 
the  relative  positions  of  the  plate  and  tube.  Take  out  the 
ground  glass,  and  put  in  one  with  a  transparent  film.  Focus 


ENGINEERING  SURVEYING 


357 


on  this,  and  make  another  mark.  In  actual  work  this 
difference  must  be  allowed  for  by  changing  the  focus  after 
the  removal  of  the  ground  glass,  so  the  film  on  the  plate 
will  be  in  the  right  position.  The  instrument  maker  should 
see  to  the  register,  but  it  is  as  well  to  test  his  work. 

As  everything  depends  upon  the  focal  distance  this  must 
be  accurately  determined.  Lens  makers  usually  state  the 
focal  distance,  but  as  it  is  liable  to  vary,  the  operator  had 
better  determine  it  himself.  For  simple  convex  lens, 
double  or  piano  convex  lens,  measure  from  optic  center  to 
surface  of  ground  glass.  For  double  compound  lenses  pro- 
ceed as  follows  (Fig.  227) : 


6' 


FIG.  227. 


Set  up  several  stakes  in  the  ground  distant  from  0  about 
two  or  three  hundred  feet  as  SS'S"S'"S"".  With  transit 
at  0,  measure  angles  SOS",  etc.  Set  up  the  camera  at  O, 
level  it  carefully,  make  the  image  5"  coincide  with  a  ver- 
tical line  through  the  center  of  the  plate,  and  photograph 
the  stakes.  The  greater  the  distance  apart  on  the  plate 
of  the  stakes  the  more  accurate  will  be  the  determination  of 
the  focal  length.  GG'  represents  the  plate,  OP  the  focal 
length.  Measure  s"s""  on  plate,  then 


OP  = 


s"s"' 
tan  a 


SOS"  =  a. 

For  a  test  of  distortion  of  the  lens,  with  OP  just  found, 
compute  the  angles  SOS',  SOS",  etc.,  and  if  they  agree  with 
angles  taken  with  the  transit  the  lens  is  free  from  distortion. 


358  PRACTICAL   SURVEYING 

Next,  the  horizon  of  the  view  must  be  found.  Find  the 
center  of  the  ground  glass  and  draw  a  vertical  and  hori- 
zontal line  through  it.  Level  the  instrument  carefully, 
and  set  beside  it  an  engineer's  level,  with  the  telescope  at 
same  height  as  the  lens  of  the  camera.  With  the  level  find 
some  object  in  the  distance.  Turn  the  camera  to  this 
object  and  move  the  object  slide  up  and  down  until  the 
object  is  exactly  at  the  intersection  of  the  lines  on  the 
ground  glass;  the  object  slide  is  then  in  its  normal  position, 
and  a  scratch  on  the  slide  will  determine  that  position  for 
all  time.  This  scratch  should  be  marked  zero,  and  gradu- 
ations should  extend  above  and  below  it.  The  lower  grad- 
uations should  have  a  minus  sign  before  their  numbers. 
The  plate  holder  should  have  four  fine  needles,  so  inserted 
in  the  frame  that  their  shadows  will  be  photographed. 
When  the  picture  is  developed,  lines  scratched  on  the  plate 
and  connecting  these  points  will  occupy  the  same  positions 
as  the  lines  drawn  on  the  ground  glass.  The  maker  can 
fix  these  needles  in  place. 

The  horizontal  line  represents  the  horizon  of  the  picture, 
and  is  the  trace  of  a  line  on  a  level  with  the  center  of  the 
instrument.  The  object  of  graduating  the  vertical  move- 
ment of  the  object  slide  is  to  provide  for  a  changing  of  the 
horizon  when  necessary  to  limit  sky  views.  By  noting  the 
number  on  this  index  when  the  view  is  taken,  the  actual 
horizon  of  the  picture  is  set  off  from  the  horizon  of  the  instru- 
ment when  plotting. 

One  more  adjustment  and  we  are  done.  This  is  to  meas- 
ure the  field  of  view.  Half  the  length  of  the  plate,  divided 
by  the  focal  length  gives  the  tangent  of  half  the  horizontal 
angle.  The  horizontal  angle  is  the  field  of  view  we  require, 
and  dividing  360°  by  this  angle,  gives  us  the  number  of 
views  needed  to  go  around  the  circle. 

Half  the  width  of  the  plate,  divided  by  the  focal  length, 
gives  us  the  tangent  of  half  the  vertical  angle.  As  a  gen- 
eral proposition  it  may  be  stated  that  the  greater  the  focal 
length,  the  smaller  the  field  of  view  and  the  greater  the 
accuracy  in  the  work.  The  smaller  the  focal  length,  the 
greater  the  field  of  view,  greater  rapidity  (because  fewer 
views)  and  less  accuracy. 

To  take  a  view,  set  up  the  camera  and  level  it  carefully. 


ENGINEERING  SURVEYING  359 

Adjust  to  focal  length  and  set  the  object  slide  to  the  most 
favorable  position  and  note  index  number  for  fixing  horizon 
in  views.  Then  adjust  the  stop,  set  in  the  plate  holder  and 
verify  the  leveling.  The  levels  are  apt  to  get  a  little  out 
during  all  the  handling.  When  all  is  ready,  take  the  picture. 

THE   PLATTING 

In  the  test  for  distortion,  the  whole  idea  of  the  method  for 
using  the  proof  is  given.  The  proofs  are  conical  projec- 
tions, and  the  optic  center  the  point  of  view.  The  objects 
represented  are  so  far  distant  that  their  images  are  formed 
on  the  same  focal  plane,  and  the  point  of  sight  remains 
constant  as  in  perspective  drawing. 

On  the  plate  draw  the  horizontal  line  (the  horizon)  and 
the  vertical  line,  from  the  shadows  of  the  points  of  the 
needles.  If  the  objective  was  above  or  below  the  horizon, 
then  instead  of  drawing  it,  draw  a  line  parallel  to  it  above 
or  below,  as  indicated  by  the  index  number  observed. 
From  the  vertical  line  measure  to  the  right  or  left  to  the 
object  you  wish  to  locate,  and  divide  this  distance  by  the 
focal  length,  this  will  give  the  tangent  of  the  horizontal 
angle  from  the  line  of  sight.  From  the  horizontal  line, 
which  is  a  trace  of  the  plane  of  the  optic  center,  measure 
up  or  down,  as  the  case  may  be,  to  the  object,  and  divide 
this  distance  by  the  focal  length  to  obtain  the  tangent  of 
the  vertical  angle. 

Every  point  located  on  the  map  must  show  in  two  views 
at  least.  These  views  are  taken  from  points  previously 
fixed  by  the  triangulation  or  by  direct  measurement.  The 
points  from  which  the  views  are  taken  must  be  plotted,  and 
from  these  points  lines  drawn  on  the  bearings  given  in  the 
field  notes  when  the  view  was  taken.  On  these  bearings 
lay  off  the  focal  distance  and  at  the  end  of  this  line  draw 
one  at  right  angles  to  represent  the  plate.  On  the  line 
representing  the  plate,  lay  off  on  either  side  the  distances 
from  the  vertical  to  the  object,  and  from  the  point  of  view 
draw  lines  through  these  points.  The  lines  through  two 
plates  produced  to  an  intersection  locate  the  objects. 

Fig.  228  illustrates  well  the  method  of  plotting.  OOf  rep- 
resent the  points  from  which  the  views  were  taken.  GGf 


PRACTICAL  SURVEYING 


and  G"G'"  the  plates,  OP  and  O'P'  the  line  of  direction  of 
sight.  ABC,  etc.,  and  A'B'C,  etc.,  represent  on  the  plate 
the  objects  to  be  located  and  their  positions  on  the  maps  are 
shown  by  the  points  of  intersection. 


FIG.  228. 


To  fax  the  elevation,  measure  the  distance  from  point  of 
sight  to  object,  and  multiply  into  tangent  of  vertical  angle 
already  found ;  add  to  the  elevation  of  the  point  from  which 
the  sight  was  taken,  the  height  of  instrument,  and  add 
or  subtract,  according  to  whether  the  point  is  above  or 
below  the  horizon,  the  height  above  ascertained.  This  will 
give  the  elevation  of  the  object  above  datum. 

Spherical  aberration  does  not  interfere  with  the  accuracy 
of  the  work,  provided  the  focal  length  is  ascertained  by 
means  of  a  point  near  the  extremity  of  the  plate  in  the 
horizon.  (See  Fig.  227.) 

FIELD  WORK 

The  field  work  may  be  performed  in  one  of  three  ways, 
or  a  combination  of  all. 

i.  The  ground  may  be  triangulated  with  the  transit 
and  views  taken  from  the  triangulation  points  with  the 
camera,  the  direction  of  the  views  to  be  ascertained  by 
azimuths  from  the  lines  between  stations.  These  azimuths 
are  to  be  taken  by  a  compass,  or  by  means  of  a  horizontal 
limb. 


ENGINEERING  SURVEYING 


2.  The  camera  may  be  used  in  connection  with  a  pocket 
compass,  the  work  starting  from  a  measured  base. 

3.  The  work  may  be  done  with  a  camera  alone,  fitted 
with  a  compass,  horizontal  limb  and  vertical  circle.      In 
this  case  the  triangulation  is  carried  on  with  the  work. 
With  ordinary  care  either  method  is  good.     The  camera 
must  be  rigid  and  the  plate  truly  vertical. 

Below  is  a  form  of  record. 


View. 

Station. 

Number. 

Index  number. 

Bearing. 

Remarks. 

A  

I 

O 

1951° 

2 

O 

227° 

B  

I 

—  2 

84° 

The  index  number  may  be  the  same  or  different  for  all 
views  at  the  same  station.  Any  time  of  the  year  is  good  for 
this  work,  and  any  hour  of  the  day  when  the  air  is  clear. 
Long  distances  between  stations  should  be  chosen  as  short 
bases  increase  error.  A  few  views  only  are  necessary,  as 
sketches  may  be  made  of  unimportant  places,  and  these 
views  should  be  well  chosen.  A  little  care  exercised  in 
selecting  positions  will  save  much  office  work. 

OFFICE  WORK 

Upon  the  scale  used  depends  the  accuracy  of  the  plotting. 
If  the  scale  is  large  then  very  long  sights  should  not  be 
attempted,  but  if  the  scale  is  small  then  of  course  the  range 
can  be  longer.  The  error  in  height  is  in  proportion  to  the 
distance,  and  Professor  Hardy  says  that  with  a  focal  length 
of  1.64  ft.  this  error  will  not  exceed  I  ft.  in  550  yds. 
Colonel  Laussedat  has  shown  that  with  a  scale  of  -3-^-$  with 
a  focal  length  of  0.5  m.  points,  1500  meters  distant,  may  be 
represented,  and  with  a  scale  of  ^TrW  the  operations  could 
be  conducted  at  4000  meters. 

When  the  plates  are  prepared  for  plotting  the  office  notes 
are  placed  in  a  book  in  seven  columns  as  follows : 


362  PRACTICAL  SURVEYING 

FORM  OF  RECORD  TO  REDUCE  HEIGHT  TO  COMMON  DATUM 


View. 

Distance. 

Point. 

Ref.  ft. 

Ref.  of  sta., 

ft. 

True  elev., 
ft. 

Remarks. 

Va. 

I 

-(-IO2 

-{-460 

+  562 

2 

+  QO 

55° 

—     2< 

4.7IT 

The  first  column  is  for  the  views. 

The  second  column  contains  the  distances  to  the  points. 

The  third  column  the  names  or  numbers  of  the  points. 

The  fourth  column  the  height  above  or  below  station. 

The  fifth  column  elevation  of  station. 

The  sixth  column  elevation  of  point. 

The  seventh  column  for  remarks. 

For  drawing  in  contours  the  fixing  of  natural  and  artificial 
objects  on  the  plan,  with  their  heights  noted,  will  give  all 
the  data  necessary  together  with  a  close  inspection  of  the 
proofs  as  the  work  proceeds.  In  the  case  of  a  bare  country, 
with  no  buildings,  fences  or  trees,  a  few  painted  stakes  or 
flags  put  in  at  salient  points  will  serve. 

It  is  best  to  work  directly  from  the  negatives  as  the  paper 
positives  are  too  much  affected  by  atmospheric  changes. 
Blue  prints  are  as  easy  to  work  from  as  silver  prints  if  posi- 
tives are  used. 

Surveying  by  camera  has  equal  advantages  with  survey- 
ing by  plane-table  as  it  is  a  graphic  process,  but  it  is  more 
exact  than  the  plane-table  as  atmospheric  changes  have  no 
effect  on  the  records.  It  is  more  rapid  in  the  field  work,  and 
is  accurate  for  more  than  one  scale  in  the  plotted  work. 

As  compared  with  transit  and  stadia  it  is  more  rapid 
in  the  field,  and  a  little  quicker  in  the  office.  Like  it,  the 
records  may  be  kept,  and  reproduced  at  any  time  to  differ- 
ent scales,  and  it  has  a  further  advantage  in  that  all  errors 
of  observation  are  entirely  eliminated.  Various  plans  for 
saving  labor  in  the  office  work  have  been  suggested,  but 
this  paper  is  already  long  enough. 

Before  closing,  attention  should  be  called  to  one  point, 
namely,  that  the  notes  may  be  sent  to  a  draftsman  who 
never  was  on  the  work,  and  a  correct  map  may  be  drawn 
by  him.  No  other  method  approaches  it  in  this  particular. 


APPENDIX   A 
•    ESSENTIALS   OF  ALGEBRA 

The  average  person  whose  mathematical  training  does  not  extend 
beyond  that  given  in  the  grade  school  arithmetic  is  afraid  of  shadows 
when  reading  technical  books. 

Many  men  declare  an  expression  like 

V  =  -Kh  (R*  -  r2) 
or 

v  =  ^k±A 

2 

to  be  algebra.  Not  having  studied  algebra  they  slur  over  their 
studies  until  finally  the  art  of  using  symbols  is  acquired  to  some 
degree  and  they  believe  they  know  enough  of  algebra  for  practical 
purposes.  What  they  have  learned  is  the  ability  to  evaluate  simple 
formulas  but  of  algebra  they  know  nothing.  The  two  expressions 
above  given  are  not  algebra  but  are  merely  arithmetical  rules  written 
in  mathematical  shorthand. 

In  ordinary  language  in  a  grade  school  arithmetic  the  rule  for 
finding  the  volume  of  a  hollow  cylinder  appears  as  follows: 

Multiply  the  radius  of  the  outside  of  the  cylinder  by  itself  (that  is, 
square  it]  and  subtract  from  the  product  the  product  of  the  radius  of  the 
inside  of  the  cylinder  by  itself.  Multiply  the  difference  between  these 
products  by  the  height  of  the  cylinder  and  this  final  product  is  to  be  multi- 
plied by  3.1416. 

Written  in  mathematical  shorthand  the  rule  appears: 

V=  irk  (R*  -  r2), 

in  which  TT  =  3.1416,  the  ratio  of  the  circumference  to  the  diameter 
(that  is  the  circumference  is  3.1416  greater  than 
the  diameter), 

363 


PRACTICAL   SURVEYING 


h  =  height  of  cylinder, 

R  =  radius  of  the  outside  of  the  cylinder, 

r  =  radius  of  the  inside  of  the  cylinder, 

V  =  volume  (cubic  contents)  of  the  cylinder. 

When  h,  R  and  r  are  in  feet,  V  =  cubic  feet.  When  h,  R  and 
r  are  in  inches,  V  =  cubic  inches.  Similarly  for  units  in  the 
metric  or  any  other  system. 

Enclosing  some  of  the  factors  in  a  parenthesis  indicates  that  they 
are  first  dealt  with,  the  result  being  a  single  factor. 

A  frustum  of  a  cylinder  is  shown  in  Fig.  229.  A  rule  for  the 
volume  as  written  in  a  school  arithmetic  is  as  follows: 

Multiply  the  area  of  the  base  by  the  average  height. 

Assuming  that  the  student  does  not  know  how  to  obtain  the 
area  of  the  base  the  rule  will  be  written  as  follows: 

Square  the  radius  of  the  bottom.  Multiply  the  product  by  the  aver- 
age height  and  this  result  by  3.1416. 

Written  in  mathematical  shorthand  it  appears : 


in  which     V  =  volume  of  frustum, 

TT  =  3.1416, 

r  =  radius  of  base, 

h\  =  height  of  one  side, 

hz  =  height  of  other  side. 


(Note.     If  the  diameter  is  used  instead  of  the  radius  divide  3.1416 
by  4  so  the  rule  appears  V  =  0.7854  d2  (  —       — -J-j 

EXAMPLES. 

i.   What  is  the  volume  of  a  ring  (cylinder)  8  ins.  long,  inside  diam- 
eter 6  ins.  and  outside  diameter  7  ins.? 

Here    h  =  8;  R  =  J;  r  =  | ;  R2  =  12.25;  r2  =  95 
V  =  3.1416  X  8  X  (12.25  -  9)  =  3-1416  X  8 
X  3.25  =  81.68  cu.  ins. 


ESSENTIALS   OF   ALGEBRA  365 

2.  What  is  the  volume  of  a  frustum  of  a  solid  cylinder  10  ins.  in 
diameter,  having  one  side  9  ins.  long  and  the  other  side  15  ins.  long? 

V  =  3.1416  X  52  X  -       — "  =  942.48  cu.  ins. 

No  matter  how  accustomed  one  may  become  to  handling  such 
expressions  the  use  of  letters  is  generally  confusing  without  some 
knowledge  of  algebra,  so,  for  the  benefit  of  students  who  experience 
difficulty  in  understanding  the  mathematical  expressions  and  using 
the  formulas  in  this  book  the  author  will  attempt  to  present  some  of 
the  essentials  of  algebra,  which  is  in  fact  only  an  extension  of 
arithmetic. 

Arithmetic,  as  defined  in  textbooks,  is  "the  science  of  numbers  and 
the  art  of  computation."  Sir  Isaac  Newton  termed  Algebra  "uni-. 
versal  arithmetic,"  for  it  enables  us  to  reason  out  in  a  general  way 
solutions  for  many  problems  which  by  ordinary  arithmetic  would 
involve  much  cut-and-try  work.  It  is  therefore  an  abbreviated 
method  for  solving  questions  relating  to  numbers  and  quantities. 
In  proceeding  to  solve  a  problem  algebraically  the  algebraic  work  is 
completed  when  the  formula  is  derived,  the  solution  of  the  problem 
from  that  point  being  done  arithmetically.  Briefly  stated  the  reason- 
ing is  done  by  algebra  and  the  work  by  arithmetic. 

All  the  fundamental  operations  of  algebra  depend  upon  the  single 
principle  that  quantity  bears  no  relation  to  order.  When  a  quantity 
is  to  be  increased,  or  diminished,  by  other  quantities  the  result  will 
be  the  same  no  matter  in  what  order  the  work  is  done,  provided  none 
of  the  quantities  be  neglected. 

WHY  LETTERS  ARE  USED  IN  ALGEBRA 

In  arithmetic  ten  characters,  o,  i,  2,  3,  4,  5,  6,  7,  8,  9,  are  used 
to  represent  numbers,  there  being  no  limit  to  the  size  of  a  number 
which  may  be  expressed  by  a  proper  arrangement  of  the  characters. 
These  characters,  or  figures,  are  in  themselves  sufficient  for  use  in 
solving  all  problems  in  common  arithmetic  except  those  in  which 
three  numbers  are  given  and  it  is  necessary  to  find  a  fourth.  The 
method  used  was  once  known  as  the  "Rule  of  Three,"  the  problems 
being  termed  problems  in  proportion. 


366  PRACTICAL  SURVEYING 

Proportion  involves  both  multiplication  and  division.  During 
part  of  the  work  some  character  must  be  used  to  represent  the  un- 
known quantity  until  it  has  a  numerical  value,  the  letter  x  being 
generally  used.  The  work  is  expressed  in  the  following  manner: 

5  :  10  ::  20  :  x, 

which  is  read,  As  5  is  to  IO,  so  is  20  to  x. 

The  two  end  quantities  are  known  as  the  extremes;  the  two  middle 
quantities  (or  terms)  as  the  means.  The  product  of  the  means  = 
the  product  of  the  extremes,  therefore, 

10   X   20  c, 

-  =  40    .-.    *  =  40. 


Algebraically  it  is  written, 

5         20 


10        x 


and  it  is  solved  as  follows: 


10 

5  #  =  10  X  20  =  200, 
therefore 

200 
x  =  -  -  =  40. 

5 

When  two  expressions  are  equal  they  are  written  with  the  sign  of 
equality  (  =  )  between  them  and  thus  a  new  complete  expression 

is  obtained  known  as  an  equation.    -^  =  —  is  a  typical  equation; 

10         x 

5/10  is  known  as  the  left  side  and  2O/X  as  the  right  side.  The 
resolution  of  an  equation  is  accomplished  by  collecting  all  the  known 
quantities  on  one  side,  the  unknown  quantity  then  standing  alone 
on  the  other  side.  Algebra  is  used  in  separating  the  known  from 
the  unknown  quantities,  after  which  the  work  is  arithmetical  and 
the  unknown  quantity  receives  a  numerical  value.  Any  analogy,  or 
proportion,  may  be  changed  into  an  equation  by  making  the  product 
of  the  extremes  equal  the  product  of  the  means. 


ESSENTIALS  OF  ALGEBRA  367 

A  knowledge  of  the  rules  and  methods  of  algebra  and  common 
arithmetic  enables  one  to  solve  equations  without  difficulty.  Given 
a  problem  the  mathematician  first  attempts  to  form  a  clear  idea  of 
its  nature  and  then  tries  to  express  its  terms  and  the  relation  of  its 
parts  in  algebraic  language,  using  the  letters  x,  y,  Z,  etc.,  for  the 
unknown  quantities.  Usually  more  time  is  spent  in  attempting  to 
express  a  problem  than  is  spent  in  solving  it. 

When  a  problem  is  proposed  in  which  there  are  several  unknowns 
there  must  be  as  many  independent  equations  as  there  are  unknowns. 
If  the  problem  is  properly  limited  the  equations  may  be  readily 
written. 

Suppose  we  have  two  quantities  x  and  y. 

Their  sum  amounts  to  18. 

This  is  written  x  +  y  =  18. 

Their  difference  is  6;  x  —  y  =     6. 

Their  product  is  72;  x  X  y     or     xy  =  72. 

x 
Their  quotient  is  2;  x  -i-  y     or     ~  =     2. 

Expressed  as  a  proportion  x  :  y  : :  4  :  2 . 

Expressed  as  an  equation  2  x  =  4  y. 

The  sum  of  their  squares  is  180;  X2  +  y2  =  180. 

The  difference  of  their  squares  is  108;  x2  —  y2  =  108. 

The  values  of  X  and  y  are  not  material,  the  above  equations  being 
illustrative  only.  The  values  are  found  by  solving  two  simultaneous 
simple  equations.  This  may  be  done  by  addition  or  subtraction. 

By  addition: 

x  +  y  =  18 
x  —  y  =    6 

2  x          =  24 

24 

X   =   -*   =    12 
2 

.*.    x  =  12     and     y  —  6. 
Here  the  —  y  and  -{-y  =  o. 


368  PRACTICAL  SURVEYING 

By  subtraction: 

x  +  y  =  18 

x  —  y  =  6 

2  y  =  12 

12 

y  =  —  =  6. 

2 

Here  the  signs  in  the  subtrahend  were  changed  from  +  to  — 
and  from  —  to  +. 

When  the  relation  of  one  unknown  quantity  to  another  is  simple, 
a  letter  may  be  used  for  one  of  them  and  an  expression  for  the  other 
deduced  from  the  relation  between  them.  This  shortens  the  work 
and,  in  the  language  of  mathematicians,  "renders  it  more  elegant." 
For  example,  if  the  difference  between  two  quantities  be  5,  let  y  be 
the  less,  then  x  =  y  +  5 ;  or,  let  x  be  the  greater  and  y  =  x  —  5. 

The  work  will  often  be  shortened  if  letters  are  taken,  not  for  the 
unknown  quantities  themselves,  but  for  their  sum,  difference  or  any 
other  relation  from  which  their  values  may  be  readily  found. 

A  competent  mathematician  does  not  consider  the  solution  of  a 
problem  to  be  complete  unless  it  exhibits  all  the  cases  which  can 
occur;  and  the  results  which  flow  from  contrary  suppositions  can 
only  be  exhibited  by  expressions  such  as  those  to  be  considered 
when  we  demonstrate  algebraic  multiplication  and  division.  In 
practical  work  algebra  is  applied  only  when  position  or  other  condi- 
tions must  be  assigned  to  quantities  which  are  frequently  to  be 
found  under  such  conditions  that  they  are  sometimes  to  be  added 
and  sometimes  to  be  subtracted.  The  use  of  the  positive  (  +  )  sign 
to  indicate  when  a  quantity  is  to  be  added,  and  the  negative  (  — ) 
sign  when  it  is  to  be  subtracted  does  not  alter  the  meaning  affixed 
to  these  signs  (that  is  to  indicate  opposite  conditions  or  states  of 
being)  in  the  algebraic  sense,  for  they  are  prefixed  solely  for  the 
purpose  of  subjecting  the  quantities  to  algebraic  processes.  This 
will  be  illustrated  when  we  touch  on  negative  exponents. 

USE   OF  SIGNS 

In  algebra  multiplication  is  indicated  by  (X)  as  in  arithmetic. 
When  quantities  are  represented  by  letters  instead  of  figures,  a  dot 


ESSENTIALS   OF   ALGEBRA  369 

is  sometimes  used,  for  the  multiplication  sign  might  be  mistaken  for 
an  x.  Thus  a  X  b  and  a  •  b  have  the  same  meaning. 

The  dot  is  not  used  between  figures  for  it  might  be  mistaken  for  a 
decimal  point,  although  in  practice  when  used  to  indicate  multi- 
plication it  is  written  in  a  higher  position  than  a  period  or 
decimal  point.  It  is  customary  to  write  letters  with  no  sign  or 
mark  between  them  when  they  are  to  be  multiplied  together,  so 
a  X  b,  a  •  b  and  ab  have  the  same  meaning.  However,  if  several 
letters  form  one  quantity  when  grouped  and  this  is  to  be  multiplied 
by  a  letter  or  another  group  of  letters,  the  dot  is  sometimes  used  to 
collect  the  factors.  For  example  abode  means  aXbXcXdXe, 
or  a  •  b  •  c  •  d  •  e,  but  abc  •  de  means  that  abc  form  one  factor  and 
de  another,  so  a  continued  product  is  not  wanted,  but  the  product 
of  two  factors. 

Division  is  represented  in  the  following  ways,  in  the  order  in 

which  they  are  favored  by  mathematicians,  7,  a/6,  a  -5-  b. 

b 

The  parenthesis  is  very  common  in  algebraic  work.  Numbers  or 
letters  enclosed  within  parentheses  are  treated  as  one  group.  For 
example  a  (b  -f-  c  -f-  d)  means  ab  +  ac  -f-  ad,  for  a  is  a  factor 
common  to  the  three  letters,  which  may  therefore  be  first  added 
together  and  the  sum  multiplied  by  a. 

The  sign  for  addition  (+)  is  termed  the  plus  sign  in  arithmetic 
and  the  sign  for  subtraction  (  — )  is  termed  the  minus  sign.  These 
two  signs  in  arithmetic  indicate  operations  to  be  performed. 

When  used  in  algebra  the  plus  sign  is  called  the  positive  sign  and 
the  minus  sign  is  called  the  negative  sign.  These  signs  do  not  indi- 
cate operations  but  represent  existing  conditions.  Whatever  is  not 
positive  must  be  negative,  and,  conversely,  whatever  is  not  negative 
must  be  positive.  x 

Addition  and  subtraction  in  algebra  are  independent  of  the  signs, 
for  negative  quantities  may  be  added  to,  or  subtracted  from,  posi- 
tive or  negative  quantities;  also  positive  quantities  may  be  sub- 
tracted from,  or  added  to,  negative  as  well  as  positive  quantities. 

The  recognition  of  the  fact  that  negative  quantities  actually  exist 
and  may  be  operated  on  as  though  they  are  positive  is  the  first 
striking  difference  between  algebra  and  common  arithmetic. 


370  PRACTICAL  SURVEYING 

The  money  a  man  possesses  is  positive.  The  money  he  owes  is 
negative.  These  are  elementary  facts  known  to  every  person.  When 
a  bookkeeper  balances  his  books  he  performs  an  operation  in  algebra, 
or,  in  mathematical  language,  "obtains  the  algebraic  sum"  of  the 
money  passing  through  his  hands. 

A  surveyor  calls  the  elevation  of  the  first  floor  of  a  building  zero, 
or  ±o  (plus  or  minus  zero).  All  heights  above  are  positive  (-f) 
and  all  depths  below  are  negative  (  — ).  From  the  first  floor  to  the 
cellar  floor  the  distance  is  9  ft.  (  —  9)  and  from  the  first  to  the  second 
floor  the  distance  is  16  ft.  (  +  16). 

Find  the  difference  in  elevation  between  the  basement  floor  and 
the  second  floor. 

Without  any  knowledge  of  algebra  a  person  would  say  9  +  16  = 
25  ft.,  which  is  correct. 

The  elevation  of  the  second  floor  is  +16  and  the  elevation  of  the 
basement  floor  is  —  9.  How  far  is  it  from  the  second  floor  to  the 
basement  floor? 

Ans.  —   9  that  is        —   9 

+  16   i  -16 

-25  -25 

The  negative  sign  prefixed  to  the  result  shows  the  measurement 
to  be  in  a  negative  direction. 
How  far  is  it  from  the  basement  floor  to  the  second  floor? 

Ans.  +16  that  is       +16 

-   9  +  9 

+  25  +25 

The  positive  sign  prefixed  to  the  result  shows  the  measurement  to 
be  in  a  positive  direction. 

In  the  first  question  the  second  floor  elevation  was  subtracted 
from  that  of  the  basement  floor.  In  the  second  question  the  base- 
ment floor  elevation  was  subtracted  from  that  of  the  second  floor. 
Thus  the  answers  show  direction  as  well  as  amount.  A  study  of 
the  work  shows  that  the  distance  between  two  points  was  found 
and  one  quantity  had  not  been  lessened  by  taking  it  from  another 
as  in  arithmetic. 


ESSENTIALS  OF  ALGEBRA  371 

The  algebraic  sum  is  the  difference  between  the  sums  of  the  posi- 
tive and  negative  quantities  or  expressions,  hence  the  rule  for  sub- 
traction in  algebra:  Change  the  sign  of  the  subtrahend  from  +  to  — , 
or  from  —  to  + ,  and  then  obtain  the  algebraic  sum. 

A  man  has  $7.00  and  on  pay  day  receives  $15.00.  He  owes  a 
friend  $1.50,  pays  $3.50  for  a  meal  ticket,  pays  $5.00  for  a  pair  of 
shoes  and  $6.00  for  a  hat.  What  has  he  left? 

Place  the  amounts  in  two  columns. 

+   7-00 
+  15-00 

+  22.00 
—  l6.00 

+  6.00 

In  the  above  example  the  process  of  subtraction  is  purely  arith- 
metical and  the  result  obtained  is  the  algebraic  sum  of  the  positive 
and  negative  sums  of  money. 

Suppose  the  man  purchased  a  suit  of  clothes  for  $18.00,  the  result 
would  be  +6.00  —  18.00  =  —$12.00,  showing  he  would  be 
in  debt  to  the  amount  of  $12.00. 

A  man  swims  up  a  stream  at  the  rate  of  4  miles  per  hour  against  a 
current  of  2  miles  per  hour.  How  far  does  he  go  in  6  hours? 

Let  his  rate  be  +4  and  the  current  be  —2,  for  the  directions  are 
opposite.  Then  in  6  hours  he  will  go6(+4  —2)  =  +12  miles. 
The  positive  sign  indicates  that  he  went  up  stream  12  miles.  Had 
the  current  been  —4  miles  and  the  swimmer  could  only  proceed  at 
the  rate  of  +2  miles  per  hour,  then  in  6  hours  he  would  find  him- 
self 12  miles  down  stream,  that  is  6  (  —  4  +  2)  =  —  12  miles. 

Letters  as  well  as  figures  are  used  freely  in  algebraic  work  to 
represent  either  quantities  or  numbers.  When  a  sign  of  operation 
is  placed  between  figures  or  letters  they  are  termed  factors.  Thus 
a,  b  and  C  are  factors  in  the  expression  abc,  or  a  X  b  X  c\  and  2,  6 
and  7  are  factors  in  the  expression  2  X  6  X  7.  Factors  are  num- 
bers from  the  multiplication  of  which  a  product  results. 

A  figure  and  letter  are  often  written  together,  as  5  a,  6  h,  7  &, 
meaning  5X0,  6  X  h,  J  X  k.  The  figure  and  letter  together  form 


372  PRACTICAL  SURVEYING 

one  factor,  each  being  a  coefficient  of  the  factor.  The  figure  is  the 
numerical  coefficient  and  the  letter  is  the  literal  coefficient  of  the  factor. 
For  the  sake  of  clearness  the  word  "coefficient"  is  used  to  indicate 
the  numerical  coefficient,  the  word  "letter"  indicating  the  literal 
coefficient.  Figures  in  arithmetic  are  characters  used  to  represent 
numbers.  The  word  "figure"  does  not  always  mean  a  number,  so 
it  is  common  to  use  the  word  "number"  in  mathematical  work  in- 
stead of  "figure,"  a  custom  to  which  the  author  will  adhere. 

ADDITION  AND   SUBTRACTION 

The  rule  for  obtaining  the  algebraic  sum  as  a  process  of  addition, 
the  rule  for  subtraction  having  been  already  given,  is  as  follows: 

Case  I.  When  the  quantities  are  alike.  If  the  signs  are  alike  add 
the  coefficients.  If  the  signs  are  unlike  take  the  difference  between  the 
coefficients.  To  the  sum  or  difference  prefix  the  sign  of  the  greater  and 
annex  the  common  letter  ,  or  letters. 

Case  II.  When  the  quantities  are  unlike.  Write  them  one  after 
another  with  their  proper  signs  and  coefficients. 

EXAMPLES. 

Addition.  3  #  ~  5  b  +  4  c  ~  3  d  —  2  e 

6  a  +  2  b  —  70  —  4d  -j-  S  e 


Subtraction.  8  ab  —  2  cd  +  $  ac  —  7  ad 
3  ab  +  4  cd  +  5  ac  —  2  ad 
$  ab  —  6cd*  -  5  ad 

When  no  sign  is  written  in  front  of  a  quantity  it  is  understood  that 
the  quantity  is  positive. 

If  a  —  b  is  to  be  added  to  3  a  we  may  first  subtract  b  from  a 
and  add  the  remainder  to  3  a;  or  subtract  b  from  3  a  and  add  a  to 
the  remainder.  The  result  in  either  case  =  4  a  —  b.  Since  a  and  b 
are  unlike  quantities  the  only  way  their  sum  can  be  represented  is 

*  In   the  example  in   subtraction   the  signs   in  the  subtrahend  were 
mentally  changed. 


ESSENTIALS  OF  ALGEBRA  373 

to  write  a  +  b',  conversely  their  difference  can  only  be  represented 
by  writing  a  —  b,  or  b  —  a.    ' 

To  subtract  a  —  c  from  3  a  we  may  first  subtract  c  from  a  and 
then  subtract  the  remainder  from  3  a\  or  add  C  to  $  a  and  then  sub- 
tract a.  Thus,  3  a  —  (a  —  c)  =  2  a  +  c.  If  a  —  c  is  to  be  sub- 
tracted from  3  a  -f-  2  c,  subtract  a  from  3  a  and  add  c,  the  remainder 
being  2  a  -f  3  c.  Thus  3  a  +  2  c  —  (a  —  c)  =  2  a  +  $  c.  In 
the  foregoing  illustrations  of  subtraction  the  work  has  all  been  done 
by  changing  the  sign  of  the  quantities  to  be  subtracted. 

A  little  thought  will  show  that  we  may  add  or  subtract  any  two 
terms,  without  regard  to  the  other  terms  with  which  they  may  be 
connected. 

DEFINITIONS 

A  term  is  a  simple  quantity;  as  a,  ab,  4  be,  etc. 

Like  terms  are  those  of  which  the  literal  parts  are  the  same;  as 
4  ab,  ab,  9  ab,  etc. 

Unlike  terms  are  those  which  consist  of  different  letters;  as  2  ab, 
3  be,  5  cd,  etc. 

Compound  quantities  consist  of  several  terms  connected  by  the 
signs  +  or  — .  A  two  term  quantity  is  called  a  binomial;  a  three 
term  quantity  a  trinomial;  a  four  term  quantity  a  quadrinomial; 
a  quantity  containing  more  than  four  terms  is  a  polynomial,  or 
multinomial.  It  is  customary  to  speak  of  all  quantities  higher  than 
binomials  as  polynomials. 

Compound  quantities  are  sometimes  enclosed  within  parentheses, 
which  indicates,  as  already  explained,  that  the  group  thus  enclosed 
must  be  first  operated  upon  before  using  the  other  factors.  When 
two  or  more  negative  signs  are  enclosed  within  a  parenthesis  it 
indicates  continued  subtraction. 

When  the  parenthesis  is  preceded  by  a  positive  sign  remove  the 
parenthesis  and  proceed  to  work  as  indicated  by  the  signs. 

When  the  parenthesis  is  preceded  by  a  negative  sign  it  means 
the  group  as  a  factor  is  to  be  subtracted.  Therefore  change  all  the 
signs  within  the  parenthesis,  the  +  to  —  and  the  —  to  + ,  after 
which  remove  the  parenthesis. 

When  several  parentheses  are  encountered  first  remove  the  inner 
one  and  work  to  the  ends. 


374  PRACTICAL  SURVEYING 

EXAMPLE.    Remove  the  parentheses  in  the  following  expression: 

5  a  -  [2  a  +  (-  3  a  -  4  b)  -  (a  -  8  b)  +  4  a] 
=  5  a  —  (2  a  —  30,  —  46  —  a  +  8  £  +  4  a) 
=  5  a  -  (2  a  +  4  b) 
=  $  a  —  2  a  —  4  £ 
=  30-46. 

MULTIPLICATION 

In  algebraic  work  the  signs  of  the  quantities  must  always  be 
considered.  When  multiplying  algebraic  quantities  like  signs  pro- 
duce +  and  unlike  signs  — .  That  is,  +a  X  —a  —  —  a2; 
aa  =  a2  and  —  a  X  —  a  =  a2. 

Multiply  the  coefficients  and  to  the  product  annex  the  letters  of  both 
factors.  If  the  multiplicand  is  compound  multiply  each  term  sepa- 
rately by  the  multiplier. 

EXAMPLE. 

$  a  —  4  6  +  3  c  —  2  d  +  e  —  i 

5<* 

25  a2  —  20  ab  +  15  ac  —  10  ad  +  5  ae  —  $  a 

If  the  multiplier  is  compound  multiply  first  by  one  of  its  terms,  then 
by  another,  etc.,  and  add  the  products. 
EXAMPLE. 

2  x2  —  3  xy   +6 

3  x2  +  3  xy   —  5 

6  x*  -  9  tfy  +  18  x2 

+  6  opy  —  9  x2y2  +  18  xy 

-  lox2 +  i$xy  -  30 

6  x4  —  3  x*y  +  8  x2      -  9  x*y*  +  33  xy  —  30 

It  helps  if  the  quantities  are  arranged  in  some  order.  In  the  ex- 
ample given  above  the  product  comes  out  according  to  the  descend- 
ing values  of  X}  both  multiplicand  and  multiplier  having  been  so 


ESSENTIALS  OF  ALGEBRA  375 

arranged.  This  carries  out  the  principle  that  quantity  is  indepen- 
dent of  order  and  that  the  positive  and  negative  signs  indicate  a  con- 
dition or  state  of  being  rather  than  an  operation. 

Powers  of  the  same  quantity  are  multiplied  by  adding  their  expo- 
nents, this  fact  being  dealt  with  in  the  chapter  where  the  use  of  loga- 
rithms is  explained. 

Multiplication  is  a  shortened  method  of  addition  as  division  is  a 
shortened  method  of  subtraction.  In  multiplying  a  —  b  by  C,  we 
may  either  first  subtract  and  then  multiply,  or  first  multiply  and 
then  subtract.  The  latter  is  the  order  in  algebra;  we  first  multiply 
a  by  C  =  ac,  and  then  b  by  c  =  be.  The  latter  is  subtracted 
from  the  former,  the  result  being  ac  —  be,  with  the  signs  the  same 
as  those  of  the  multiplicand. 

In  multiplying  a  —  b  by  c  —  d,  first  multiply  a  —  b  by  C  as 
before  =  ac  —  bc\  then  multiply  a  —  b  by  d  =  ad  —  bd,  which 
subtract  from  the  first  product.  This  is  accomplished  by  changing 
the  signs,  when  it  becomes  —  ad  +  bd,  the  signs  being  the  opposite 
of  those  in  the  multiplicand. 

The  first  and  last  terms  show  that  quantities  with  like  signs  mul- 
tiplied together  produce  +  ,  and  the  other  two  terms  show  that 
those  which  have  unlike  signs  produce  — . 

DIVISION 

When  the  signs  are  alike  the  sign  of  the  quotient  is  +,  but  if  the 
signs  be  unlike  the  sign  of  the  quotient  is  — . 

The  above  statement  is  self-evident;  for  the  divisor  multiplied 
by  the  quotient  must  produce  the  dividend  with  its  proper  sign.' 
The  whole  operation  depends  upon  the  principle  that  the  value 
of  a  quantity  is  not  altered  by  both  multiplying  and  dividing  it  by 
the  same  quantity. 

Powers  of  the  same  quantity  are  divided  by  subtracting  the  ex- 
ponent of  the  divisor  from  that  of  the  dividend;  the  remainder  is 
the  exponent  of  the  quotient.  See  the  chapter  dealing  with  loga- 
rithms. 

When  the  divisor  is  a  simple  quantity,  write  it  under  the  dividend  in 
the  form  of  a  fraction;  cancel  like  quantities,  and  divide  the  coefficients 
by  their  greatest  common  divisor. 


376  PRACTICAL  SURVEYING 

EXAMPLES. 


//  the  dividend  is  compound  divide  each  term  separately  by  the  divisor. 
EXAMPLE. 

6^  a3b2(?  —  4.2  a2b3(?       o  ac 

-  —  2    _  i.  be 

14  a2b2c2  2 


g  divisor  is  compound  arrange  the  terms  of  the  dividend  and 
divisor  according  to  the  powers  of  some  letter.  Divide  the  first  term  oj 
the  dividend  by  the  first  term  of  the  divisor  to  obtain  the  first  term  of  the 
quotient,  then  multiply  the  whole  divisor  by  this  term,  and  subtract  the 
product  from  the  dividend;  bring  down  as  many  terms  to  the  remainder  as 
may  be  requisite  for  a  new  partial  dividend,  with  which  proceed  as  before. 
EXAMPLE. 


a3  - 
a3  - 

3 

a2b 

+.3 

+   2 

ab2  -  V 
ab2 

a 

-  b 

+ 

b2 

a2 

—  2  ab 

— 

2 
2 

a2b 
a2b 

+     afc2  -  W 
+     a62  -  ft8 

Note  that  while  the  signs  are  written  in  the  partial  subtrahends 
as  produced  by  the  multiplication,  they  are  changed,  mentally,  for 
each  subtraction. 

When  in  dividing  the  last  remainder  is  a  simple  quantity,  place 
the  divisor  below  it  in  the  form  of  a  fraction,  and  annex  it  with  its 
proper  sign  to  the  quotient. 

FRACTIONS 

Fractions  are  the  same  in  algebra  as  in  common  arithmetic  save 
for  the  fact  that  letters  are  used  in  algebra.  They  may  be  added, 


ESSENTIALS  OF  ALGEBRA  377 

subtracted,  multiplied  and  divided  by  the  rules  already  given  for 
dealing  with  whole  quantities.  They  are  reduced  to  a  common 
denominator  exactly  as  in  arithmetic. 

NEGATIVE   QUANTITIES 

A  negative  quantity  standing  by  itself  has,  strictly  speaking,  no 
meaning. 

For  example  if  C  be  the  difference  between  a  and  b  the  algebraic 
expression  is  a  —  b  =  c,  when  a  is  greater  than  b',  or  b  —  a  = 
—  c,  when  a  is  smaller  than  b.  The  expression  —  c,  however,  is 
impossible  arithmetically,  for  a  greater  quantity  cannot  be  taken 
from  a  less. 

Join  the  negative  quantity  to  another,  as  m  —  C  and  the  expres- 
sion may  be  subjected  to  all  the  operations  of  algebra,  therefore  it 
is  proper.  If  there  is  any  absurdity  it  does  not  appear  until  the 
result  is  obtained,  when  it  shows  that  some  condition  inconsistent 
with  the  other  conditions  has  been  admitted  into  the  question.  A 
negative  result  when  it  agrees  with  the  steps  of  the  processes  in 
solving  a  problem  is  a  proper  algebraic  result,  for  it  points  out  the 
impossibility  of  the  conditions  and  thus  has  a  use,  in  that  it  places  a 
limit  on  the  terms  of  the  question. 

In  all  algebraic  work  it  is  necessary  to  pay  close  attention  to  the 
positive  and  negative  expressions  and  the  forms  which  result  from 
them.  There  must  be  no  hesitation  in  the  operations  for  it  has  been 
shown  how  quantities,  and  single  terms  may  be  added,  subtracted, 
multiplied  and  divided  by  one  another  and  how  the  signs  of  the 
results  are  obtained.  These  signs  do  not  belong  to  the  terms  taken 
as  isolated  quantities,  but  express  the  relation  existing  between  the 
terms  of  the  result. 

The  product  of  a  —  b  by  a  —  b  =  a2  -  -  2  ab  -f  b2  and  the 
product  of  b  —  a  by  b  —  a  =  a2  —  2  ab  +  b2.  There  is  nothing 
in  the  product  to  indicate  whether  a  is  greater  or  less  than  6,  or, 
looking  at  it  in  another  way,  .if  a  —  b  =  c,  whether  the  product 
has  arisen  from  -\-c  or  from  —  c,  for  each  of  these  squared  =  -\-C2. 
The  square  root  of  -\-c2  may  be  either  -\-c  or  —  c  and  the  square 
root  of  —  c2  is  impossible. 


378  PRACTICAL  SURVEYING 

The  formula  a2  —  b2  =  (a  -f  b)  (a  —  b)  is  of  great  service  in 
algebra  and  should  be  memorized.    Let 


a2  +  b2  =  a2  -  b2  X  -i  =  (a  +  b  Vi)  (a  -  b  VT). 

The  use  of  negative  exponents  often  simplifies  work.    We  can 

ab 

divide  ab  by  a2  by  writing  them  in  the  form  of  a  fraction  —  and 

a2 

a*       a2 

canceling    like   quantities,  —  --  -  =   a3-!,  or  merely  subtract  the 
a2       a2 

exponent  of  the  divisor  from  that  of  the  dividend  a5"2  =  a3. 
Now  divide  a2  by  a5.     By  writing  the  work  in  fractional  form 

CL  T  T 

•—  =  —  .    By  subtraction,  a2  —  ab  =  a"3  '     so  that  a"3  =  —  . 
a5       a3  a3 

In  such  examples  the  negative  exponent  does  not  represent  a  nega- 
tive quantity,  but  only  shows  that  the  quantity  placed  in  the  numer- 
ator should  be  in  the  denominator.  In  either  place  it  can  be  subjected 
to  all  the  rules  of  algebra,  but  in  some  problems  the  use  of  the  nega- 
tive exponent  is  a  convenient  method  for  getting  rid  of  fractions. 
The  examples  given  show  that  any  quantity  may  be  removed  from 
the  numerator  to  the  denominator,  or  vice  versa,  by  changing  the 

nj  9 

sign  of  the  exponent.    Thus  -  -  =  a?bc~2    and    ab~2c2  =  -—  • 

c*  cr 

Roots  may  be  expressed  by  fractional  exponents,  as  V  a  =  a2, 
and  fractions  by  negative  exponents  as  shown,  so  the  following 
expressions  should  be  carefully  studied: 


xa 


In  arithmetic  it  is  often  found  to  be  convenient  to  reduce  a  com- 
mon fraction  to  a  decimal,  and  the  negative  exponent  of  a  fraction 
is  the  result  of  an  equivalent  operation. 


ESSENTIALS  OF  ALGEBRA  379 

Notice  the  ease  with  which  the  following  expressions  are  handled 
by  the  use  of  fractional  and  negative  exponents. 

Vx  X  ^x  =  #M  =  x*x*  =  #M  =  a* 
^  _-*-*_  xl  -  Vx 


Exponents,  the  student  must  remember,  are  added  and  subtracted, 
being  logarithms. 

SECONDARY   OPERATIONS 

To  facilitate  work  a  number  of  rules  have  been  evolved  from  a 
study  of  the  development  of  algebraic  expressions  when  raised  regu- 
larly to  higher  powers.  An  expert  algebraist  is  thus  enabled  to  save 
a  great  deal  of  time  because  he  remembers  how  one  factor  follows 
another.  He  not  only  can  write  quickly  without  actual  multiplica- 
tion, the  expansion  of  an  expression  to  any  degree,  but  conversely 
can  "factor"  expressions  of  any  degree.  By  factoring  is  meant  to 
reduce  an  expression  to  its  elemental  factors. 

EQUATIONS 

A  statement  in  algebraic  language  that  one  quantity  equals  an- 
other is  an  equation. 

A  simple  equation  is  one  of  the  first  degree.  That  is,  the  radical 
sign  and  exponents  are  not  used  in  either  the  statement  or  the  solu- 
tion. 

A  quadratic  equation  is  of  the  second  degree  and  the  square  root 
of  a  quantity  appearing  in  it  must  be  obtained. 

A  cubic  equation  is  of  the  third  degree;  a  bi-quadratic  equation  is 
of  the  fourth  degree,  etc. 

The  degree  of  an  equation  is  the  degree  of  its  highest  term. 

To  solve  an  equation  the  following  rules  must  be  observed  in  the 
order  given. 


380  PRACTICAL  SURVEYING 

ist.    Clear  of  fractions. 

If  any  term  be  divided  by  any  quantity,  multiply  every  term  by  this 
divisor  (denominator} . 

2nd.  Write  all  the  known  quantities  on  one  side  of  the  equation 
and  the  unknown  quantities  on  the  other  side.  Then  collect  by 
addition. 

Any  term  may  be  transposed  from  one  side  of  an  equation  to  the  other 
by  changing  its  sign  from  +  to  — ,  or  from  —  to  -\-. 

If  a  term  be  found  on  both  sides  with  the  same  sign  it  should  be  can- 
celed in  both. 

yrd.  If  the  unknown  quantity  be  multiplied  by  any  other,  divide 
both  sides  by  the  multiplier. 

In  this  way  the  value  of  an  unknown  quantity  is  found  when  there 
are  no  surds  or  powers. 

4th.  If  the  equation  contains  a  surd  (an  expression  of  a  root  of  an 
algebraic  quantity  which  is  not  a  complete  power)  bring  the  surd  to  one 
side  by  itself,  take  away  the  radical  sign  and  raise  the  other  side  to  the 
corresponding  power. 

5//f.  //  one  side  of  the  equation  be  a  complete  power  take  the  corre- 
sponding root  of  both  sides. 

At  no  time  during  the  process  of  solving  an  equation  can  the 
sides  be  unequal  and  the  foregoing  rules  maintain  equality  of  the 
sides  as  follows: 

Rule  i.    Both  sides  are  multiplied  by  the  same  quantity. 

Rule  2.  In  transposition  the  same  quantity  is  subtracted  from 
both  sides. 

Rule  3.     Both  sides  are  divided  by  the  same  quantity. 

Rule  4.    Both  sides  are  raised  to  the  same  power. 

Rule  5.    The  same  root  is  taken  of  both  sides. 

Solve  2  x  —  —  =  3_*  -h  4- 

4          4 

Clear  of  fractions,  8^—19  =  3^  +  16. 

Transpose,  8  X  —  3  x  =  16  +  19. 

Collect,  5  x  =  35. 

Divide  by  5,  x  =  ^  =  7. 


ESSENTIALS   OF  ALGEBRA  381 

Solve,  (3  x  -f  i)*  +  5  =  10. 

Transpose,  (3  x  -f-  i)2  =  10  —  5  =  5. 

Square  by  Rule  4,  3  x  +  I   =  52  =  25. 

Transpose,  3^  =  25  —  i   =  24. 

Divide  by  3,  x  =  —  =  8. 

o 

Solve,  9  x2  +  9  =  3  x2  +  63. 

Transpose,  9  X2  -  3  x2  =  63  -  9  =  54. 

Collect,  6x2  =  54. 

Divide  by  6,  x2  =  *r  =  9. 

Extract  the  root,  X  =  Vq  =  3. 

It  has  been  stated  that  any  proportion  or  analogy  may  be  turned 
into  an  equation  by  making  the  product  of  the  first  and  fourth  terms 
equal  to  the  product  of  the  second  and  third  terms.  This  will  now  be 
illustrated. 

2  +  x  :  6  —  x  ::  15  : 9. 

Then  9  (2  +  x)  =  15  (6  -  x). 

or  18  +  9  x  =  90  —  15  x. 

Transpose,  9  X  +  15  x  =  90  —  18  =  72. 

Collect,  24  x  =  72. 

Divide  by  24,  x  =         =  3. 

x  —  5  :  2  x  ::  5  :  20. 
Then  20  X  x  —  5  =  5X22; 

or  20  x  —  100  =  10  x. 

Transpose,          20  x  —  10  X  =  100. 
Collect,  IOX  =  100. 

IOO 

Divide  by  10.  X  =  =  10. 

10 

The  age  of  a  man  is  a  and  the  age  of  his  son  is  b.  In  how  many 
years  will  the  father  be  n  times  as  old  as  the  son? 


382  PRACTICAL  SURVEYING 

Solution.  As  time  passes  there  will  be  added  to  each  age  an  incre- 
ment, x.  The  ratio  n  between  the  ages  of  the  father  and  son  de- 
pends upon  this  factor,  or  increment.  Then 


_ 


Transpose,  a  -f  x  =  nb  +  nx. 

Collect,  nx  —  x  =  a  —  nb. 

x  (n  —  i)  =  a  —  nb. 

a  —  nb 
Divide,  x 


n  —  i 

Assume  the  father's  age  to  be  26  and  that  of  the  son  to  be  4.    When 
will  the  father  be  twice  as  old  as  the  son? 

26  -  (2  X  4) 
x  =  s 3*  =  18  years. 

2    —    1 

a  +  x  _  26  +  18       44 
b  +  x  "     4  +  18   "~  22  " 

Problems.  Answers. 

I-    5*  +  3  =  2X  +  I5-  x  =  4. 

62     vV  x- 

—  x  =  4  — x  =  6. 

o 

^      x   .   x  _  _3_ 

3'    2       3       4  "  9i3 

x2  a 

4.   ^  —  a  =  '.  x  =  — 

x  -  a  2 

s.  _^_+^  =  6.  ,,  =  /6-2 

T        I       /y  T     /y 

T^    A/  1  wV 


6.   x  +  («"  +  r»)*  = 


(a2  +  *2)*  V^ 

Equations  may  also  be  solved  by  factoring. 


ESSENTIALS  OF  ALGEBRA  383 

SIMULTANEOUS   SIMPLE  EQUATIONS 

When  there  are  several  unknown  quantities  there  must  be  an  in- 
dependent equation  for  each  quantity;  from  these  an  equation  must 
be  deduced  which  contains  only  one  of  the  unknowns  and  this  equa- 
tion may  be  solved  by  the  preceding  rules.  Three  rules  are  in  gen- 
eral use  for  solving  simultaneous  simple  equations.  It  is  immaterial 
which  is  used.  The  object  is  to  eliminate  one  unknown  after  another 
until  but  one  is  left. 

Rule  I.    This  appears  to  be  the  most  regular. 

(a)  Find  a  value  for  one  of  the  unknowns,  assuming  all  the  rest  to 
be  known. 

(b)  Make  these  values  equal  to  one  another  and  from  them  find  a 
value  for  another  unknown. 

(c)  Repeat  (b)  successively  with  the  remaining  unknowns  until  the 
final  equation  results. 

Rule  II.  This  is  a  little  shorter  than  the  first,  but  the  reductions 
are  more  intricate. 

(a)  Find  a  value  for  one  of  the  unknown  quantities  in  that  equation 
in  which  it  is  the  least  involved. 

(b)  Substitute  this  value  and  its  powers  for  that  unknown  quantity 
and  its  powers  in  all  the  other  equations. 

(c)  Proceed  as  in  (b)  with  these  equations  to  get  rid  of  other  unknown 
quantities. 

Rule  III.     This  is  the  most  simple  and  expeditious. 

(a)  Multiply  the  equations  by  such  quantities  as  will  make  the  co- 
efficients of  one  of  the  unknown  quantities,  or  of  its  highest  power,  the 
same  in  all  the  equations. 

(b)  If  these  equal  terms  have  like  signs  subtract  them,  but  if  signs  are 
unlike  add  them,  thus  giving  rise  to  new  equations. 

(c)  Continue  as  in  (a)  and  (b)  with  the  new  equations. 
Examples  under  Rule  III. 

i.  Let  the  equations  'be  I2  | .     Find  x  and  y. 

5*  +  3?  =  50) 
First  equation  multiplied  by  5,     5  #  +  5  ;y  =  60 

5  *  +  3  y  =  SQ 

2  y  =  10 


3*4 


PRACTICAL  SURVEYING 


Divide, 


IO 

y---s. 

x  =  12  —  5 


*  +     y  +    ^  =    53 
2.  Let  the  equations  be  x  +  2y  +  $z  =  105 

#  +  3:v  +  4z  =  134 

Second  equation,  x  +  2  y  -}-  $  z  =  105 
First  equation,    x  X      y  +      2  =     53 
Subtract, 


.  Find  X,  y  and  2. 


y  +  2Z  =     52=  Fourth  equation. 


Third  equation,    #. 
Second  equation,  ^ 
Subtract, 

Fourth  equation, 
Fifth  equation, 


42;  =  134 
32  =  105 


y  -f 


29  =  Fifth  equation. 


y  -f-  2  Z  =     52 

y  +      z  =     29 

a  =     23 


y   =   29   -  2   =    29   -   23    =   6. 

*  =  53  ~  (6  +  23)  =  24. 

By  substituting  these  values  in  the  original  equations  they  may  be 
solved  and  a  proof  obtained  of  the  accuracy  of  the  work. 


I. 


2.     - 
2 


Problems. 

-  $y  =     90. 

+  $y  =  160. 

*  =z6. 


x      y 

Z   =    2. 

5       9 


Answers, 
x  =  30. 
y  =  20. 

x  =  20. 
?  =  18. 


a. 


3-   x  +  y 

x2  -  y>  -  b. 


2  a 


2  a 


ESSENTIALS  OF  ALGEBRA  385 

3#  -  U  =  2X  +  y  +  i  x  =  13. 

3  5  y  =    3- 

The  student  must  remember  that  the  solution  of  equations  in- 
volving fractions  is  as  easy  as  the  solution  of  equations  in  which  no 
fractions  appear.  They  merely  take  a  little  more  time,  and  care 
must  be  exercised  that  nothing  is  missed.  It  has  been  stated  before 
that  fractions  in  algebra  are  treated  exactly  as  fractions  are  treated 
in  common  arithmetic. 

I 

5.   x  +  100  =  y  -f  z,  x  =  9  — 


II 

b  +  a  —  c 

2 
a  +  c  —  b 

2 
c  -f  b  —  a 


z  +  100  =  3  x  +  3  y.  z  =  63 

6.       x  +  y  =  a.  a 

a;  +  z  =  b.  y 

y  +  z  =  c.  i 


7. 

I 
-  x 

+  % 

.    i 

+  -z 

=    62. 

2 

3 

4 

I 

i 

i 

2 

4 

5* 

-47- 

I 

4 

i 
-\-  -  v 

5 

+;• 

=  38. 

#  =  24. 
»  =  60. 

2=    I2O. 

In  problems  containing  fractions  the  work  is  simplified  by  recast- 

T  y 

ing  some  of  the  expressions.     For  example  -  x  may  be   written  - 

2  2 

,2  2  # 

and  -  x  may  be  written  — ,  etc. 
o  X 


386  PRACTICAL  SURVEYING 

QUADRATIC  EQUATIONS 

A  quadratic  equation  often  comes  to  the  engineer  in  the  following 
form 

M  =  Rbh2, 

then  h=\/M 


Rb' 

in  which  M  =  resisting  moment  (should  be  equal  to  the  bending 

moment), 

R  =  moment  factor  (unit  moment  of  resistance), 
b  =  breadth  of  beam, 
h  =  height  (depth)  of  beam. 

For  a  rectangular  beam  the  resisting  moment  =  •*—  -   in  which 

6 

/  =  maximum  fiber  stress  (skin  stress)  and 


A  pure  quadratic  equation  contains  only  the  second  power  of  the 
unknown  quantity. 

An  affected  quadratic  equation  contains  the  first  and  second 
powers  of  the  unknown  quantity. 

Affected  quadratic  equations  assume  one  of  the  following  three 
forms: 


i. 


x2  +  ax  =  +b 


.      ,.  .  -a  ±  Va2  +  46 

in  which  x  =  • 

2 

2.  x2  —  ax  =  +b 

+a  ±  Va2  +  4  b 
in  which  x  =  —  • 

2 

3.  x2  —  ax  =  —  b 

•      i..  i.                              +a  ±  Va2  -  4  b 
in  which  x  =  ~ — 


ESSENTIALS  OF  ALGEBRA  387 

The  student  should  commit  the  above  formulas  to  memory. 

Since  the  square  root  of  x2  —  2  ax  +  a2  is  either  a  —  x  or  x  —  a, 
the  root  of  the  known  side  of  the  equation  must  have  both  the  signs 
+  and  —  before  it.  Sometimes  both  give  the  proper  solutions, 
and  at  other  times  only  one  of  them  is  proper. 

If  a  positive  answer  is  required  the  sign  of  the  radical  in  (i)  and 
(2)  must  be  -f-,  but  in  (3)  it  may  be  either  -f-  or  —  .  There  is  how- 
ever a  limitation  in  this  case  for  4  b  must  not  be  greater  than  a2, 
otherwise  the  quantity  below  the  radical  sign  would  be  negative 
and  its  root  impossible.  This  happens  when  the  absolute  term  b  is 

greater  than  -  a2,  the  square  of  -  the  coefficient  of  x. 
4  2 

In  solving  an  affected  quadratic  several  methods  are  at  the  service 
of  the  mathematician.  Some  problems  are  more  readily  solved  by 
one  method  than  by  another.  The  method  best  retained  by  the 
memory  is  that  of  completing  the  square.  The  methods  in  the 
order  of  general  custom  are  as  follows: 

i.   By  inspection. 

The  sum  of  the  roots  is  equal  to  the  coefficient  of  the  first  power 
of  the  unknown  quantity  with  the  sign  changed,  and  the  product  of 
the  roots  is  equal  to  the  right-hand  member  of  the  equation  with  the 
sign  changed. 

x2  —  6  x  =  72. 

Sum  of  the  roots,  =  -f-  6. 

Product  of  the  roots,       =   —  72. 


2.  By  factoring. 

x2  —  6  x  =  72. 

x2  —  6  x  —  72  =  p. 

(x  +  6)  (x  —  12)  =  o. 

x  =  12. 

3.  By  the  rule. 

x2  -  6x  =  72. 


_  ±  6  ±  V62  +  4  X  72 

«v    —  ~ 


388  PRACTICAL  SURVEYING 

4.   By  completing  the  square. 

The  student  should  work  the  following  by  actual  multiplication, 
observe  the  formation  carefully  and  memorize  the  results. 

(a  +  b)  (a  +  b)  =  (a  +  b)2  =  a2  +  2  ab  +  b\ 
(a  -  b)  (a  -  b)  =  (a  -  b)2  =  a2  -  2  ab  +  62. 
(a  -  b)  (a  +  b)  =  a2  -  b2. 

To  solve  a  quadratic  by  completing  the  square  add  to  both  sides 
the  square  of  half  the  coefficient  of  the  unknown  quantity. 

x2  —  6  x  -f  9  =  72+9  =  81. 

In  the  above  line  the  square  of  one-half  of  6  was  added  to  both 
sides  of  the  equation. 

Extract  the  square  root  of  both  sides. 

*  -  3  =  9- 
Solve  for  x. 

x  =  9  +  3  =  I2- 
Sometimes  the  square  of  the  unknown  quantity  has  a  coefficient: 

3#2  -  6x  =  72, 

in  which  case  the  equation  must  first  be  divided  by  this  coefficient; 
x2  —  2  x  =  14. 

To  avoid  fractions  multiply  the  equation  by  4  times  the  coefficient 
of  x2  and  then  add  the  square  of  the  original  coefficient  of  x. 

Let  the  equation  be  7  x2  —  20  x  —  32. 

Multiply  by  4  X  7  =  28,    196  x2  —  560  x  —  896. 
Add  202  =  400, 

196  x2  —  $60  x  -f  400  =  896  +  400  =  1296. 
Extract  the  root,  14  x  —  20  =  ±36. 

X  =   +4,     or      -!-• 

If  an  equation  contains  two  powers  of  the  unknown  quantity  and 
the  exponent  of  one  is  double  that  of  the  other  it  may  be  solved 
like  a  quadratic. 


ESSENTIALS  OF  ALGEBRA  389 

Let  the  equation  be  x&  —  6  x?  =  16. 

Completing  the  square,  X6  —  6^  +  9  =  16  +  9  =  25. 

Extracting  the  root,  r*  —  3  =  ±  $. 

Transposing,  X?  =  3  d=  5  =  8. 

Extracting  the  cube  root,  X  =  2. 

Problems.  Answers. 

i.    8  +  x2  -  6  =  80. 


2.     -    +   42  -    = h    20- 


b  -  a)*. 

(18  +  e  ±  6  Ve  +  9)"' 

7.  A  brick  pier  30  ins.  wide  and  42  ins.  long  carries  a  load  of 
450,000  pounds.  Design  a  foundation  of  the  same  shape  resting  on 
earth  having  a  safe  bearing  value  of  3000  pounds  per  square  foot. 
What  should  be  the  length  and  breadth  of  the  footing? 

4150,000 

Required  area  =  :iii-J =150  sq.  ft. 

3000 

Ratio  of  sides  =  --  =  1.4. 
30 

Let  X  =  width, 

then  1.4  x  =  length. 

x  (1.4*)  =  150- 
1.4  x2  =  150. 


x 

1.4 

1.4  x  =  1.4  X  10.35  =  r4-45 
Make  the  footing          14'  6"  X  ior  4". 


390  PRACTICAL  SURVEYING 

8.  Find  a  number  of  which  the  square  shall  be  4  times  the  num- 
ber together  with  5  times  the  same  number.  Ans.  g. 

g.  Find  two  numbers  of  which  the  sum  is  133  and  their  difference 
is  47.  Ans.  go  and  43. 

10.  Find  two  numbers  in  the  ratio  of  4  to  3,  so  that  if  i  be  added 
to  each  of  them,  the  sums  shall  be  in  the  ratio  of  9  to  7. 

Ans.  8  and  6. 

11.  Divide  72  into  two  parts  so  that  three  times  the  greater  shall 
exceed  twice  the  less  by  121.  Ans.  53  and  19. 

12.  A  general  sends  1500  of  his  troops  to  the  West  and  J  of  his 
entire  army  to  the  East.     He  retains  i  of  the  army  after  detaching 
1200  to  the  rear  guard.     How  many  men  are  in  the  army? 

Ans.  16,200. 

13.  A  and  B  started  out  in  life  with  equal  amounts  of  money. 
A  spent  annually  $600  more  than  his  income,  while  B  saved  $800 
annually.     In  12  years  B  was  twice  as  rich  as  A.     What  was  their 
original  capital? 

(x  -  60  X  12)   =  x  +  80  X  12. 

Ans.  $24,000  each. 

14.  An  aeroplane  going  at  the  rate  of  90  miles  an  hour  overtakes 
another  having  a  start  of  180  miles,  but  going  at  the  rate  of  70  miles 
an  hour.     How  many  hours  did  it  require? 

Ans.  g  hours. 

15.  A  cistern  can  be  filled  with  water  by  one  faucet  in  12  hours 
and  by  another  in  8  hours.     In  what  time  will  it  be  filled  if  both 
run  together?  Ans.  43  hours. 

1 6.  Two  men  can  together  do  a  piece  of  work  in  8  days.    One  can 
do  it  alone  in  12  days.     How  long  would  it  take  the  other? 

Ans.  24  days. 

17.  A  can  do  a  piece  of  work  in  50  hours;  B  in  60  hours  and  C  in 
75  hours.     In  what  time  can  the  three  do  it  when  working  together? 

Ans.  20  hours. 

18.  A  and  B  together  can  do  a  piece  of  work  in  12  hours;  A  and  C 
together  in  20  hours;   B  and  C  together  in  15  hours.     In  what  time 


ESSENTIALS   OF   ALGEBRA  391 

will  they  do  it,  all  working  together,  and  in  what  time  will  each  do 
it  separately? 

xxx 

-  H 1 =  2. 

12  20          15 

Ans.  Together  in  10  hours;    A  alone  in  30  hours;    B  alone  in  20 
hours;  C  alone  in  60  hours. 

19.  The  sum  of  two  numbers  =  100  and  their  product  =  2100. 
What  are  the  numbers?  Ans.  70  and  30. 

20.  Find  two  numbers  of  which  the  difference  is  8  and  the  product 
is  240.  Ans.  20  and  12. 

21.  Find  two  numbers  of  which  the  product  is  108  and  the  sum  of 
their  squares  is  225.  Ans.  12  and  9. 

22.  An  oblong  pond  was  surrounded  by  a  road  7  yards  wide,  the 
area  of  the  pond  being  15,000  square  yards  and  the  area  of  the  road 
being  3696  square  yards.     Give  the  length  and  breadth  of  the  pond. 

xy  =  15,000,     and     14  x  -h  14  y  +  196  =  3696. 

Ans.  100  X  150  yards. 

23.  Find  two  numbers  of  which  the  sum  is  13  and  the  sum  of  their 
cubes  is  637.  Ans.  8  and  5. 

24.  Sold  a  cow  for  $24  and  gained  as  much  in  per  cent  as  the 
first  cost  in  dollars.    What  was  paid  for  the  cow? 

x2 

x  H =  24. 

100 

Ans.  $20. 

25.  A  and  B  started  at  the  same  time  for  a  place  at  a  distance  of 
150  miles.    A  travels  3  miles  an  hour  faster  than  B,  and  beats  him 
by  85  hours.     At  what  rate  did  each  travel? 

Ans.  A  9  miles  per  hour,  B  6  miles  per  hour. 

26.  A  and  B  distribute  each  $1200  among  some  poor  families.    A 
gave  to  40  more  families  than  B,  but  B  gave  $5  more  to  each  family. 
How  many  families  were  helped  by  each? 

Ans.  120  by  A,  80  by  B. 

27.  A   traveling  to   Boston,  overtook  at   the  $oth    milestone  a 
band  of  sheep  traveling  at  the  rate  of  3  miles  in  2  hours;  and  2  hours 
later  met  a  wagon  moving  at  the  rate  of  9  miles  in  4  hours.     B, 
traveling  at  the  same  rate  as  A,  overtook  the  sheep  at  the  45th 


392  PRACTICAL  SURVEYING 

milestone  and  met  the  wagon  40  minutes  before  he  came  to  the 
3ist  milestone.     Where  would  B  be  when  A  reached  Boston? 

Let  x  =  distance  between  them, 

y  =  rate  of  their  traveling  per  hour. 


27  9  3 

Ans.  x  =  25. 

y  =   9- 

Problems  involving  ratio,  proportion  and  variation  are  handled  by 
the  rules  given  for  solving  equations. 

TABLE  OF  FORMULAS 

The  accompanying  table  is  of  great  value  in  many  problems. 
Let  x  and  y  be  any  two  quantities. 

S  =  x  +  y,    their  sum, 
d  =  x  —  y,     their  difference, 
p  =  xy,  their  product, 

q  =  x/y,         their  quotient, 
Z  =  x2  +  y2,  the  sum  of  their  squares, 
D  =  x2  —  y2,  the  difference  of  their  squares. 

The  use  of  this  table  will  teach  how  to  state  a  problem  and  only  a 
few  hints  will  be  here  given  as  to  its  use.  The  student  is  advised  to 
consult  it  when  called  upon  to  solve  problems. 

EXAMPLES. 

1.  The  sum  of  two  numbers  is  277  and  their  difference  is  115. 
Find  the  greater. 

From  the  table     X  -  f-^)  =  ^  +  "*)  =  196. 

2.  The  difference  of  two  numbers  is  10  and  the  product  is  no. 
Find  the  greater. 


ESSENTIALS  OF  ALGEBRA 


393 


+        " 

to 


? 


I  + 


-^,1  W 


-      1  -^,1 


+       1  + 


1  -CXJ 


" 


N 


394  PRACTICAL  SURVEYING 

From  the  table 

d  +  (d2  +  4  />)*  _  io  +  (IPO  +  476)* 

2  2 


IO  +    ^576    _      IO   +  "24 

X 


3.   The  sum  of  the  squares  of  two  numbers  is  250  and  the  differ- 
ence of  the  squares  is  88.     Find  the  numbers. 


X  = 


,  =  VsT  =  Q. 


4.  Given  the  sum  s  of  the  products  of  two  quantities,  by  known 
multipliers  m  and  n,  and  also  the  sum  of  their  products  c  by  other 
known  multipliers  p  and  q;  to  find  the  quantities. 

Here  mx  +  ny=  s,  and  px  -\-  qy  =  c;  multiplying  the  first 
equation  by  p,  and  the  second  by  m,  they  become  pmx-\-  pny  =  ps, 
and  mpx  X  mqy  =  me;  subtracting  we  get  npy  —  mqy  = 
ps  —  me ;  and  dividing  by  np  —  mq,  we  obtain 

_  ps  —  me 
np  —  mq 

In  like  manner  we  find 

(75  —  nc 
mq  —  np 

5.  Given  the  sum  5*  of  the  quotients  of  two  quantities  by  known 
divisors  m  and  n,  and  also  the  sum  c  of  their  quotients  by  other 
known  divisors  p  and  q',  to  find  the  quantities. 

X          "V  X        "V 

Here  -  +  z  =  s,     and      -  +  -  =  C, 

m       n  P       % 


ESSENTIALS   OF   ALGEBRA  395 

then  nx  +  ny  =  mns,     and     qx  +  py  =  pqc\ 

which,  resolved  as  in  (4),  gives 

pm  (ns  —  qc) 
x  —  "  ) 

pn  —  qm 

_  nq  (ms  —  pc] 
qm  —  pn 

Equations  may  be  solved  graphically  and  an  interesting  book  on 
the  subject  is  entitled,  "A  Graphical  Method  for  Solving  Certain 
Algebraic  Equations,"  by  Prof.  G.  L.  Vose,  in  Van  Nostrand's 
Science  Series,  50  cts. 

The  foregoing  rapid  presentation  of  the  essentials  of  algebra  gives 
a  student  all  that  is  necessary  in  order  to  have  a  good  working  tool. 
If  his  interest  is  aroused  and  he  desires  to  proceed  farther  in  the 
study  of  mathematics  he  should  purchase  the  text  on  algebra  used 
in  the  nearest  high  school  and  thus  when  hard  places  are  encoun- 
tered assistance  may  readily  be  obtained.  Correspondence  courses 
in  mathematics  are  given  by  the  University  of  Chicago,  Chicago,  111. 


INDEX 


Adjustments  of  instruments: 

compass,  106. 

dumpy  level,  80. 

transit,  223. 

wye  level,  79. 
Algebra,  essentials  of,  363. 
Algebraic,  formulas,  table  of,  393. 

theorems  of  sun,  163. 
Altitudes,  equal,  for  true  meridian, 

288. 
Angle,  checking  with  needle,  234. 

circular  measure  of,  169. 

denned,  118. 

supplement  of,  118. 
Angles,  deflection,  118. 

double  centering  of,  229. 

exterior,  118. 

from  bearings,  118. 

included,  118. 

logarithmic    functions   of,    190, 
203. 

natural  functions  of,  157,  194. 

repeating,  229. 

reading,  with  transit,  228. 

tables  of  functions  of,  194,  203. 
Angular  leveling,  245. 
Arc,  length  of,  latitude  and  longi- 
tude, 275. 
Area,  by  planimeter,  49. 

by  weighing,  49. 

table  of  units  of,  72. 
Areas,  computing,  128. 


J 


Areas,  correcting,  20. 

of  surfaces,  42. 
Astronomy,  268. 
Attraction,  local,  in,  118. 
Azimuth,  denned,  236.  ^ 

from  bearings,  234. 

table  of  errors  in,  276. 

Backsight,  89. 
Barlows'  tables,  58. 
Batter  boards,  54. 
Bearings,  from  angles,  118. 

from  azimuth,  234. 

platting,  124. 
Bench  marks,  89. 
Boundary  surveys,  law  of,  310. 
Bounds,  whipping  the,  6. 
Broken  line,  defined,  2. 
Bubbles,  sensitive,  importance  of, 

79- 
Building,  staking  out  of,  53. 

Care,  of  compass,  108. 

of  level  and  transit,  226. 
Carpenters'  level,  75. 
Chain,  surveying,  14. 

Gunter's,  14. 
Chaining  methods,  7. 
Chain  men,  necessity  for  good,  6. 
Chart,  Isogonic,  102. 
Checking,  levels,  91. 

rod  reading,  86. 


397 


INDEX 


Checking,  with  needle,  234. 

work,  174. 

Circular  measure  of  angle,  169. 
Compass,  surveying,  100. 

adjustments  of,  106. 

care  of,  108. 

reading  the,  104. 

use  of,  109. 
Computation,  of  areas,  128. 

of  traverses,  172. 
Contact  rods,  16. 


Contours,  341. 


- 


Corners,  establishment  of,  3,  4. 

extinct,  290. 
Crelle's  tables,  61. 
Cross,  surveyors',  36. 

wires,  78. 

Cubes,  use  of  table  of,  55. 
Curvature  of  earth,  74,  82. 
Curved  line,  definition  of,  I. 

Datum  plane,  82. 

Daylight  observations  on  Polaris, 

286. 

Declination,  100,  102,  271,  276. 
Deflection,  angles,  118. 

of  tape,  1 8. 
Deflections,    platting    angles    by, 

124. 

Departure,  defined,  128. 
Differential  leveling,  88. 
Dioptra,  210. 

Distributing  errors,  125,  133. 
Diurnal  variation,  103. 
Dividing  tables,  61. 
Double   centering   of   angles,    37, 

229. 

Double  meridian  rule,  131.     \ 
Drafting  tools,  23. 
Dumpy  level  and  adjustments,  79, 

80. 


Earthwork     computations     from 

contour  map,  344. 
Elevations,  positive  and  negative, 

83-    _ 

Ephemeris,  defined,  268. 
Equations,  solution  of,  379. 
Error,  index,  231. 

limit  of,  5. 

probable,  230. 
Errors,  in  leveling,  91. 

in  measuring,  18. 

distributing,  125,  133. 
Exponents,  56. 
Exterior  angles,  118. 
Extinct  corners,  290. 

Factors,  55. 

Foresight,  89. 

Functions  of  angles,  151,  157. 

tables  of,  194,  203. 

graphical  natural,  159. 

logarithmic,  190,  203. 
Functions,  judicial,  of  surveyors, 
293- 

Geodetic  surveying,  2. 
Geometry,  practical,  23. 
Government  surveys,  327. 
Grade  stakes,  to  set,  95. 
Grades,  selecting,  on  contours,  342. 
Gradienter  on  transit,  214. 
Graduations  on  transit,  218. 
Gravity  and  gravitation,  defined,  I. 
Greek  letters,  289. 
Gunter's  chain,  14. 

Hand  level,  94. 
Horizontal  lines,  I. 
Hubs  on  surveyed  lines,  9. 
Hydrographic  surveys,  335. 
Hypothenuse  of  triangle,  58. 


INDEX 


399 


Included  angles,  118. 

Index  error,  231. 

Intersection  method  for  surveys, 

241. 

Invar  tape,  17. 
Isogonic  lines  and  chart,  102. 

Jacob  staff,  36. 

Judicial,    the,    functions   of    sur- 
veyors, 293. 

Latitude,  128,  273. 

length  of  arc  of,  275. 
Law  of  boundary  surveys,  310. 
Legal  knowledge  of  surveyors,  5. 
Lot  surveys,  261. 
Level,  adjustment  of,  80. 

bubbles  to  be  sensitive,  79. 

carpenters',  75. 

care  of,  226. 

circular,  37. 

dumpy,  79,  80. 

for  rod,  85. 

hand,  94. 

line,  I,  73. 

Precision,  79. 

rods,  83. 

tape,  10. 

true  and  apparent,  73. 

water  tube,  77. 

wye,  79. 
Leveling,  73. 

angular,  245. 

checking  results  of,  91. 

differential,  88. 

errors  in,  91. 

profile,  or  route,  92. 

the  tape,  10. 

turning  points,  86,  89. 

with  rubber  hose,  77. 

with  transit,  216. 


Length,  tables  of  units  of,  72. 
Letters,     why    used    in    algebra, 

365. 

Limit  of  error  in  surveys,  5. 
Lines,  definitions  of,  I,  73. 

offset,  1 1 6. 

random,  22,  122. 

ranging,  21. 

Local  attraction,  in,  118. 
Locating  objects,  41. 
Logarithms,  182. 

tables  of,  201. 
Logarithmic   functions  of  angles, 

190,  203. 
Longitude,  270. 

length  of  arc  of,  275. 

Map  making,  42,  122,  261,  266. 
Marks,  bench,  for  levels,  89. 
Mean  polar  distance,  286. 
Measurement,  errors  in,  18. 
Measures,  table  of  units  of,  72. 
Measuring  on  slopes,  10. 
Mensuration,  plane,  42. 
Meridian,  determination  of,   115, 

277,  284,  288. 
Meridiograph,  281. 
Miners'  triangle,  74. 
Mining  surveys,  330. 
Missing,  lengths  and  bearings,  136, 

175- 

corners,  4. 
Multiplying  tables,  61. 

Notes  of  surveys,  II,  115. 
Negative  and  positive  elevations, 

83- 
Natural    graphical    functions    of 

angles,  159. 
Needle  reading  to  check  angles, 

234- 


400 


INDEX 


Objects,  locating,  on  survey,  41. 

Oblique  triangles,  solution  of,  165. 

Obstacles  on  line,  41. 

Offset  lines,  116. 

Omissions   in   measurements   and 

bearings,  136,  175. 
Original  surveys,  4. 

Paper,  profile,  94. 
Peg  readings,  91. 
Percenter  for  leveling,  76. 
Personal  equation,  91. 
Phototopographic  surveys,  355. 
Pins,  use  of,  in  surveying,  8. 
Plane,  defined,  2. 

surveying,  2. 

mensuration,  43. 

table,  345. 
Planimeter,  49. 
Platting,  bearings,  124. 

by  deflections,  124. 

latitude  and  departure,  266. 

stadia  notes,  353. 
Plotting,  see  platting.     Also  map 

making. 
Plats,  or  maps,  of  chain  surveys, 

42. 

Plumb-bobs,  best  weight  for,  19. 
Plunging  a  grade,  96. 
Points,  defined,  I. 

turning,  in  leveling,  86,  89. 
Polar  distance,  286. 
Polaris,  observations  on,  284,  286. 
Poles  and  rods,  21. 
Positive  and  negative  elevations, 

83. 

Practical  geometry,  23. 
Precision  level,  79. 
Probable  error,  230. 
Profile,  leveling,  92. 

paper,  94. 
Protractors,  kinds  of,  128. 


Protractors,  use  of,  123. 
Pythagorean  theorem,  149. 

Quadratic  equations,  386. 

Radian  measure  of  angle,  169. 
Radiation     method    for    surveys, 

117,  241. 

Random  lines,  22,  122. 
Ranging  lines,  21. 
Ratios,  trigonometric,  152,  153. 
Reading,  the  compass,  104. 

rods,  85. 

Refraction,  73,  82. 
Repeating  angles,  229. 
Reproducing  maps,  288. 
Resurveys,  4,  291. 
Road  percenter,  76. 
Rod,  level,  85. 

waving  the,  85. 
Rodmen,  instructions  for,  84. 
Rods,  and  poles,  21. 

contact,  1 6. 

level,  83. 

reading  of,  85. 
Roots,  square  and  cube,  55. 
Route,  leveling,  92. 

surveys,  9,  92,  339. 
Rubber  hose  for  leveling,  77. 

Sag  in  tape,  effect  of,  17. 
Scales  for  map  making,  24. 
Self-reading  rods,  83. 
Sensitive  bubbles,  79. 
Sewer  grade  stakes,  96. 
Signals  for  surveyors,  97. 
Signs,  trigonometric  ratios,  155, 

use  of,  in  algebra,  368. 
Simpson's  rule,  47. 
Slide  rule,  60. 

Slopes,  measuring  on,  7,  10. 
Smoley's  tables,  150. 


INDEX 


401 


Solar  compass,  100. 

observations,  277. 
Squares,  tables  of,  55. 
Stadia  surveys,  246,  251,  346. 
Stakes  on  surveys,  9,  52,  95. 
Stations  on  lines,  9,  n. 
Straight  line,  I. 
Straight-edge  for  leveling,  75. 
Supplement  of  angle,  118. 
Surfaces,  2. 
Survey,  notes,  n,  115. 

by  radiation,  117. 

with  transit,  233. 
Surveying,  chain,  14. 

denned,  2. 

transit,  216. 
Surveyors,  chain,  14. 

cross,  36. 

the  judicial  functions  of,  293. 

signals,  97. 
Surveys,  various  kinds  of,  defined, 

3- 

government,  327. 
hydrographic,  335. 
mining,  330. 
phototopographic,  355. 
route,  339. 
stadia,  346. 

Tables,  Barlow's,  58. 

Boileau's,  176. 

Crelle's,  61. 

Gurden's,  176. 

Lewis  &  Caunt's,  176. 

Smoley's,  150. 
Tables  of,  algebraic  functions,  393. 

functions  of  angles,  194,  203. 

lengths  of  arcs  of  latitude,  275. 

logarithms,  201. 

measures  length  and  area,  72. 

squares,  cubes,  etc.,  62. 

stadia  reduction,  251. 


Tables  of  traverse,  142. 

Tape  level,  10. 

Tapes,  1 6. 

Targets  on  rods,  83. 

Temperature,  effect  of,  17. 

Theodolite,  211. 

Tidal  level  as  datum  plane,  82. 

Time,  270. 

Transit,  212. 

surveying,  210,  222,  233. 
.   adjustment  of,  223. 

care  of,  226. 

reading  angles  with,  228. 
Traverse,  computation  of,  172. 

methods,  241. 

table  for  compass,  142. 

table  for  transit,  177. 
Trapezoidal  rule,  47. 
Triangles,  miners',  74. 

solution  of,  58,  149,  161,  165. 
Trigonometry,  148. 
Trigonometer,  161. 
Trigonometric,  functions,  151. 

laws,  164. 

ratios,  152,  153. 
Turning  points,  86,  89. 

Variation  of  needle,  100,  102. 

see  Declination. 
Verniers,  for  angles,  218. 

on  rods,  84. 
Vertical  lines,  I. 

Water-tube  level,  77. 
Waving  the  rod,  85. 
Weighing  to  obtain  area,  49. 
Whipping  the  bounds,  6. 
Wires,  in  telescope,  78. 

stadia,  246. 
Witness  stakes,  9. 
Work,  staking  out,  52; 
Wye  level,  79. 


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